Illusioners hold items horizontally in their offhand. This appears to be the case because their entire left arm is rotated incorrectly.
To reproduce
/summon minecraft:illusioner ~ ~ ~ {LeftHanded:1b,HandItems:[{id:"minecraft:bow",Count:1b}]}
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This is probably intentional I think. The offhand is position differently.
It looks differently for other entities, like players. It's rotated by 90° for the illusioner.

@[Helper] DrownedZombie
You can do a bow-shooting action… Then you may find what goes wrong.
Anyway, I feel this is not really comfortable.
You can summon a specific Pillager and compare with Illusioners.

Well I'm not sure whether U'll fix this since U had already said that Illusioners Rn't the formal mob in the game. If so, that's pretty petty :<.
Can confirm in 20w51a.
Can confirm in 1.18.2 and 22w12a.
[media][media]You can use the following command to reproduce this issue.
/summon minecraft:illusioner ~ ~ ~ {LeftHanded:1b,HandItems:[{id:"minecraft:bow",Count:1b}]}

Actually seems the item, in comparison to the arm, is correct, it's the whole arm that's rotated (look at wherethe sleeve ends, and compare it to the other arm).
Can confirm in 1.19.
Can confirm in 1.19.2.

Can confirm in 1.20.1
@unknown, please check the affected versions of the report before commenting. 🙂 I already added "1.20.1" as an affected version around five minutes prior to your comment.