I read this approx 1990 but I think it was in an anthology with a lot of stories from the 1960's and 1970's.
A man steps into his daily transporter booth and sits down, a few seconds later he goes to leave but the door won't open. While he waits for assistance he realises his head is on upside down and he starts crying, this is awkward because his nose overflows and runs down into his eyes.
A medical team arrive and take him to a holding area, there are others there who have had transportation mishaps, somebody with back to front legs and (I think) a woman with her eyes on the side of her head.
It is explained to him that he's one of the very rare casualties in the booth system, the only chance is to keep transporting him until he is OK. He notices a grotesque figure with multiple arms and legs entwined, this was a courting couple who purposely crammed into a booth together so they could be as one.
I think his first repair hop almost succeeds, his head etc is OK but his nose is still upside down, therefore he'll have to beam again.
I really can't remember what happens next.