Following Deane Yang, the answer is a definite `yes' : the map 
in question is a global diffeo,
provided that 
(a) it is `locally invertible': i.e. its   derivative is everywhere invertible,  and
(b) the domain and range are compact, simply connected, without boundary.

Proof: the map must be a covering map (``stack of records theorem'' -- see
for example Guillemin and Pollack). But a covering map between simply connected spaces
is an isomorphism -- here a diffeo. 

To make this `non-contrived' take domain and range to be the two-sphere.