Not true—in the case of a punctured torus (like the shirt with joined sleeves), it's not like a sack at all!

It can still be turned inside-out, but it's a little trickier to visualize. You can imagine the hole stretching until you have a pair of identical rings glued together, then it's easy to see that you can shrink the hole the same way except in the other direction. But there's no way to make it look like a sack with holes.