1. Hail To The King, Baby.

Here's a Bleach ask for you, what do you make of the relationship between Ichigo and Grimmjow during the arrancar arc and the narrative value of their dynamic?

Grimmjow's relationship with Ichigo is one of the best written character rivalries in the series, because much like Uluiorra he's a near perfectly crafted foil and jungian shadow of Ichigo. The only difference really between his foiling between Grimmjow, and his foiling between Ulquiorra is that the confrontation between Grimmjow helps Ichigo grow, while his relationship with Ulquiorra is entirely destructive and stunts his growth for a long time after the fact.

1. Hail to the king, baby.

Grimmjow isn't just Ichigo's enemy, he practically is Ichigo under different circumstances, the similarities betwen them run so deep. My greatest evidence for this of course is how much Grimmjow's motivations and actions resemble that of Hichigo / Zangetsu, the literal mirror to Ichigo's soul made up of his repressed desires.

Grimmjow's words, reflect Hichigo's words, which reflect Ichigo's words as well. Hichigo tries to assume control of Ichigo's body, because he believes Ichigo lacks both the desire and resolve to win because he is too afraid of internal desires like Zangetsu and has suppressed them.

Grimmjow's desire to become king, reflects Hichigo's speech about the horse and the king, that if he doesn't rule over his own emotions, that if he doesn't use his power to win then he won't be king anymore and will be ruled by someone else. As much as it seems to be deviating away from Ichigo's true goal of protecting his friends to care more about winning fights then the action of protection, Ichigo also can't help or protect anyone if he is too insecure or too unwilling to fight with everything he has.

Grimmjow also much like HIchigo, only exists in the first place because he is a strong enough Gillian that he retained his personality by consuming all of the other Gillian around him, and was able to maintain himself. He is similiar to Hichigo, and the hollow part of Ichigo's soul in general, suddenly born in emptiness given intelligence and the only way to maintain that intelligence is to fight. A smaller part of a greater whole, threatening to consume Ichigo from the inside out if he does not rule him. If Grimmjow does not fight and eat other hollows, he'll just stop existing. If Ichigo is too weak and dies, then HIchigo will stop existing as well, which is why he will hijack the body in crisis situations and soemtimes even believes he could do a better job fighting than Ichigo could.

Grimmjow and Hichigo both desire to become king, to dominate others through their power, and it's not just a selfish desire for power that fuels this, they need to fight and win in order to survive. This is also not an entirely selfish desire on Grimmjow's part, because for instance Ichigo also wants to win. The only way he can protect his friends is to gain strength and win, otherwise he's useless. Hollows are apparently empty creatures without emotions and yet Grimmjow was the king over subordinates and apparently a good enough leader that they were all willing to sacrifice their lives for him.

Which is the precise moment that Grimmjow becomes a darker reflection for Ichigo's character, because Ichigo may desire to win, he may desire to be king, but that selfish desire is also balanced by the selflessness that he mainly uses that power for the others around him. Contrast this to Grimmjow who having lost those subjects by having consumed everything around him, he has no real reason to be king.

Having no connection to the world around him, Grimmjow seeks to die by Ichigo's blade rather than just suffer the humiliation of the loss to him, because fighting is literally the only thing he has to give his life worth and meaning.

Each Hollow represents an aspect / means of death, and Grimmjow's death is by destruction. Despite his desire to be king and ruler, he has effectively lived his life (Unlife?) by destroying everything around him until there is nothing left, and there is no end to that path except self-destuction.

When Ichigo stops him despite the fact they are mortal enemies, despite the fact Soul Reapers live to fight Hollows (though, they don't kill them, quincies do but killing hollows and eradiacting their existence has always been bad, Soul Reapers have a duty to cleanse them so they can return to the cycle of souls), it's a healthy move on Ichigo's part because he's telling both Grimmjow and himself that there's more to life than just battle, and that living to fight another day is more important than dying out of shame for the loss.

2. Howling

Grimmjow functions as a jungian shadow to Ichigo as much as Zagnetsu does, he is just externalized rather than internalized. There's a rich dirge of symbolism to how Grimmjow represents the primal, repressed part of Ichigo's mind that he does not see.

To explain what a Jungian Shadow is briefly.

Jung regarded the shadow as unconscious—id and biography—suppressed under the superego's ego-ideal.[12] The shadow is projected onto one's social environment as cognitive distortions. Contrary to a Freudian definition of shadow, the idea can include everything outside the light of consciousness and may be positive or negative. Because a subject can repress awareness or conceal self-threatening aspects of the self, consensus of the idea of the shadow that it is a negative function in the self, despite the extent of the repression failing to prohibit these aspects.[16] There are positive aspects that can remain hidden in one's shadow—especially in people with low self-esteem, anxieties, and false beliefs—with these aspects being brought to the conscious mind and exercised through analysis and therapy. https://en.wikipedia.org/wiki/Shadow_(psychology)

Ichigo is incredibly repressed, his struggle throughout the series is to try to make sense and find balance within the several different parts of his soul, his quincy self, his soul reaper self, his human self, his hollow self. Zangetsu similiarly not simply Ichigo's evil side, but rather a mix of the traits he has repressed both good and bad. For example, as much as Zangetsu sometimes threatens to hijack Ichigo's body, he also does so out of love because Ichigo cares more about protecting others whereas the one Zangetsu wants to protect most of all is Ichigo.

The shadow isn't just repressed desires though, it's also theorized by Jung to be constructed of our instincts.

 Jung construed [...] the personal shadow, [as] a biological and biographical shadow unique to each person, consisting of whatever innate instincts and transpersonal potentials we have suppressed in the course of adapting to society, along with archaic and traumatic memories [of the unconscious]. [...] The personal shadow is rooted in the shadow of our social group, which has moulded our ego-ideal and world view[.]  Humphrey, Caroline (2015). "Shadows Along the Spiritual Pathway". 

Instinct, is not only what Zangetsu says that Ichigo is lacking and why he can't master his hollow self or be the king.

Grimmjow is also, literally an animal. He's a big kitty. Hollows rip off their masks to take more human forms, and then when they release their swords return to the shape of the animals they began as. Grimmjow's motivations are almost pure animal instinct, fight, kill, eat, survive.

There is so much jungian symbolism in bleach it isn't even funny, Tarot is made up of Jungian ideas, and the tarot card that has the most in common with a jungian shadow is the moon.

One of the biggest symbols of the moon card is the dual image of the dog and the wolf, howling up at the moon. The dog is the domesticated animal, the wolf the undomesticated primal instinct. The dog the disciplined part of our conscious mind we are in controll of, the wolf the unconscious mind made up of what we cannot see and cannot control.

The Moon represents your fears and illusions and often comes out when you are projecting fear into your present and your future, based on your past experiences. You may have a painful memory that caused emotional distress, and rather than dealing with the emotions you pushed them down deep into your subconscious. Now, these emotions are making a reappearance, and you may find yourself under their influence on a conscious or subconscious level. 

And, what does Zangetsu translate to?

Zangetsu (斬月, Slaying Moon) is the manifested spirit of Ichigo Kurosaki's Zanpakutō as well as his inner Hollow. Getsuga Tenshō (月牙天衝, Moon Fang Heaven-Piercer; Viz "Moon-Fang Piercer of the Heavens") is a Zanpakutō technique of Ichigo Kurosaki's Zangetsu and Isshin Kurosaki's Engetsu.

Ichigo's zanpakuto is literally named after the fact that he uses his literal repressed shadow in order to fight against his opponent, and Zangetsu takes the form more often than not, of Ichigo's reflection. Ichigo's internal world is literally an empty city over an ocean where half the city is above the ocean and half is bellow it. It's not subtle, ya'll.

Grimmjow represents how those internal fears and anxieties are a part of Ichigo. They are one in the same the domesticated dog, and the wolf. Ichigo is the king as a man, Grimmjow is the king of beasts. As I said they even have the same desire, Ichigo's desire to win is for the sake of bringing his comrades home, whereas Grimmjow's desire to win is inspired by the fact his comrades believed in him so much they were literally willing to lay down their lives and let him eat them.

Grimmjow reflects all those aspects positive and negative, he is also just in general, an ID character.

The id is the animal part of the personality, an unconscious drive to have lots of sex, survive, and thrive. It urges you to push in and eat your weight in cake. The ego is where the conscious mind lives. It's lumbered with the tricky job of satisfying the id's wild desires in a realistic and socially acceptable way.

Hunger is the word for Grimmjow, his hollow hole is literally in his stomach, his tragic backstory is he had to kill and eat his own allies, he's modeled after a panther which is a predator at the top of the food chain. However, these are also all things Ichigo needs to survive. Hichigo says as much, he needs an insatiable hunger, a ravenous bloodlust... and the id may seem selfish but if we don't eat we can't live. Just like how Ichigo's way of defeating Zangetsu is to admit to those desires that both Zangetsu and Grimmjow represents. He is once again, stabbed straight through the stomach, the same place where Grimmjow's hole is.

The only thing that gives him the strength is his desire to have his sword back and not have it get taken away from him, Zaraki of all people appears in his mind and tells him that he has a very selfish desire to keep on getting into fights and getting stronger (much like Grimmjow) however it's not entirely negative because once again Ichigo needs this killer instinct withint himself in order to win, because winning is living.

Ichigo's greatest moment of character grwoth in the fight comes when he uses his strength not only to protect Orihime, but to proect himself, the same way Hichigo does not want him to die, Orihime does not want to see him hurt, and that moment strikes Ichigo's growth.

Which is once again just how positive Ichigo and Grimmjow's relationship is, that it's only by facing Grimmjow Ichigo is pushed to learn these things about himself, only by accepting someone like Grimmjow can he accept the more negative aspects of his own self and grow past them rather than succumbing to him.