  We develop an approach to calculating the cup and cap products on Hochschild
cohomology and homology of curved algebras associated with polynomials and
their finite abelian symmetry groups. For polynomials with isolated critical
points, the approach yields a complete description of the products. We also
reformulate the result for the corresponding categories of equivariant matrix
factorizations. In an Appendix written jointly with Alexey Basalaev, we apply
the formulas to calculate the Hochschild cohomology of a simple but non-trivial
class of so-called invertible LG orbifold models. The resulting algebras turn
out to be isomorphic to what has already appeared in the literature on LG
mirror symmetry under the name of twisted or orbifolded Milnor/Jacobian
algebras. We conjecture that this holds true for all invertible LG models. In
the second part of the Appendix, the formulas are applied to a different class
of LG orbifolds which have appeared in the context of homological mirror
symmetry for varieties of general type as mirror partners of surfaces of genus
2 and higher. In combination with a homological mirror symmetry theorem for the
surfaces, our calculation yields a new proof of the fact that the Hochschild
cohomology of the Fukaya category of a surface is isomorphic, as an algebra, to
the cohomology of the surface.
