reverse, inside, palm, sole, rear, lining, wrong side
裡 centers on the hidden or opposite side of something: the inside, the rear, the lining, or the wrong side. The palm and sole senses are specific applications of this inner/underside idea.
裡 combines 衤 (clothing) with 里, which likely contributes the sound. The idea of the inside or lining of clothing extends to the general sense of inner or reverse side.
The clothing radical 衤 beside 里 suggests the inner lining of a garment. Picture turning a coat inside out to see its reverse side, the hidden part.
For リ, imagine reaching inside the lining of a coat and pulling out a small reed (リード, riido) that was hidden there: reed -> リ, and the reed was on the reverse side.