wry mouth
喎 centers on a mouth that is twisted or crooked, describing a wry or distorted mouth shape.
喎 combines 口 (mouth) with 咼, which likely contributes the sound and the idea of crookedness or distortion. The exact historical development is uncertain.
The left side is a mouth 口, and the right side 咼 looks like a twisted, uneven shape. Picture a mouth pulled into a wry, crooked grin.
For カイ, imagine a wry mouth saying 'kai' in a distorted way, like a crooked smile that makes the sound come out twisted: kai -> カイ.