Is it grammatically incorrect to say 'a shoe pair', or 'a shoes pair', instead of 'a pair of shoes?'
If it's OK, can I say 'two shoe pairs' or 'two shoes pairs' instead of 'two pairs of shoes'?
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No, I'm afraid you can't. The order can't really be changed; "a pair of" is a fixed expression.
Your only option is
A pair of shoes / Two pairs of shoes
If the word describing a pair is an adjective, then it can be used similarly to in your question.
(Game of "pairs") A correct pair
Lastly, I would mention that we can use a possessive pronoun + pair to refer to one of the items in a pair.
(I've lost a shoe - just one) Has anyone seen its pair?
You cannot use the phrase shoe pair instead of a pair of shoes.
You can say a shoe polish, shoe brush, shoe shop, etc. In these phrases, the noun shoe functions as an adjective modifying polish, brush, and shop. However, the pair is a noun which cannot be modified by any noun including the shoe. So you always say a pair of shoes, a pair of trousers, a pair of glasses, not a shoe(s) pair, a trouser(s) pair, and a glass(es) pair,