The tensor product R⊗S of two algebras can have its multiplication deformed by a bicharacter
to yield a twisted tensor product R ⊗t S. We completely describe the Hochschild cohomology
of R ⊗t S in terms of the Hochschild cohomology of the components R and S, including
the full Gerstenhaber algebra structure. This description generalizes a result of Bergh and
Oppermann. A number of interesting classes of noncommutative algebras arise as bicharacter
twisted tensor products, sometimes in non-obvious ways. The main result thereby allows us to
significantly simplify various calculations in the literature, and to compute Hochschild cohomology in several new classes of examples. In particular, we fully compute the Hochschild
cohomology of quantum complete intersection algebras, with any number of indeterminates.
One new tool which goes into the main theorem is orbit Hochschild cohomology, which can
be defined for algebras with a group action, and which satisfies twisted versions of the usual
Gerstenhaber algebra axioms.