u/Acrobatic_Caramel_74

Shoutout to Eve(@eve_firestone) from our Discord Server for the analysis!

Rhinedottir

Rhinedottir’s piece is a unicorn, a symbol of purity and grace, with her main colour being gold (just like her title). The horn of a unicorn is sometimes described as a symbol of life cycle. That lines up with Rhindottir’s wishes of creating life.

There’s also a hood? a veil? similar to the one we see her usually wearing.

The following is quoted from the description of Cinnabar Spindle (Shadow Amidst Snowstorms event weapon):

“You [Albedo] must pursue that which your elder brother, the one-horned white horse, could not accomplish. Reach the far side of philosophy, and create a new destiny for myself [Rhinedottir] and your brothers...”

So there’s this one-horned horse, too…
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Hroptatyr

Hroptatyr’s chess piece is a comedy mask. A mask is just a mask, and what lies beneath is a different story.

If we believe Pierro’s symbol to be a tragedy mask (according to Fatui constellation wheel), this may say something about their connection. They were both Royal Mages, with Hroptatyr having a higher rank and being the Chief. Are they opposite to each other, then?

Hroptatyr’s crown should be the symbol of Reason - a circlet of Logos, according to artifact types.

Idk about the gemstone in the crown - it seems to be orange, so I thought of amber, which sometimes symbolises wisdom, but it could be literally anything else. It may be a colour special to him.
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Surtalogi

Surtalogi’s chess piece is practically the same greatsword we see him wielding (shadow inside All-Devouring Narwhal). It’s sitting inside stone, so it reminds me of Excalibur story somewhat. To be fair, “Surtalogi” is literally a sword’s name (just like “Dainsleif”, “Mistilteinn” and “Skofnung”). Of course, this symbolises his pursuit of strength. In the sword, there’s the black hole symbol - like the Lightless Eye of Maelstrom, indicating his connection to Space. Purple is our usual colour of the Abyss. The sword seems to be slightly asymmetrical, compared to other chess pieces, so perhaps this points to Surtalogi being a defier of rules and order (as mentioned by Skirk).
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Rerir

Rerir’s chess piece appears to be a wolf howling at the crescent moon.

There are two pinkish-purple spikes piercing it. There’s also a similar gem in the wolf’s back. Wild Hunt summons are of the same colour.

Rerir’s polearm (according to weapon skins) is named Shattered Moon, so there’s that. The weapon he wields during his boss fight is similar to the chess piece (a crescent moon being pierced).

The wolf’s eyes are covered, and the bridle (?) seems to be pulling him up. I guess the wolf is baring teeth, too. This accentuates his blind obsession with the moon.
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And Vedrfolnir’s analysis via False Visionary(@zini1000) cuz yeah:

Vedrfolnir

Vedrfolnir’s chess piece is Ouroboros revolving around a star like a fate star but not completing the samsara I don't know what it means. Maybe he is above samsara as he has transcended it or it symbolizes that the cycle has been broken or he is escaping from it.