LEO currently has as the English translation for "etw. verkehrt herum anhaben (e.g., a T-shirt) only "to have sth. back to front".
I believe that this English is British usage. While I, as a native speaker of American English understand the expression "back to front" (it is sort of intuitive), I don't believe I would ever use it, nor have I heard other Americans using it.
If you google "He had his shirt on back to front" and "He had his shirt on backwards", you find the former has only 395 hits while the latter has over 7,720 hits.
Here's a online dictionary backing my usage of backwards for verkehrt herum in this context:
From "Dictionary.com" (entry: backwards)"
backwards or backward (ˈbækwədz)
1. towards the rear
2. with the back foremost
3. in the reverse of usual order or direction
4. to or towards the past
5. into a worse state: the patient was slipping backwards
6. towards the point of origin
7. informal bend over backwards , lean over backwards , fall over backwards to make a special effort, esp in order to please
8. informal know backwards to understand completely
As made clear from entries 2 and 3, "backwards" can be used to mean that the back is foremost, or that something is in the reverse of the usual direction (here, the shirt is on with the backside on in the reverse direction as it is supposd to be).
What is not clear to me is whether the usage of "backwards" in this context is exclusively American, or whether Brits would also use "backwards" here. Also, I am unsure of whether "back to front" should be marked as British usuage, or whether some Americans would use this as well. My intuition is that this is a chiefly or completely British expression.