It is easy to find inside out in a dictionary: 裏返し{うらがえし}, but I have difficulty translating backwards in the expression
to put a T-shirt on backwards
I believe you can use 前後ろ for "backwards" or "front-to-back". For example,
Tシャツを前後ろに着る
Edit. People in chat also point out that there is also 後ろ前 "back-to-front", so
前後ろ ↔ 前が後ろ "front to back"
後ろ前 ↔ 後ろが前 "back to front"