  We introduce a special class of convex rational polyhedral cones which allows
to construct generalized Calabi-Yau varieties of dimension $(d + 2(r-1))$,
where $r$ is a positive integer and d is the dimension of critical string vacua
with central chatge $c = 3d$. It is conjectured that the natural combinatorial
duality satisfies by these cones corresponds to the mirror involution. Using
the theory of toric varieties, we show that our conjecture includes as special
cases all already known examples of mirror pairs proposed by physicists and
agrees with previous conjectures of the authors concerning explicit
constructions of mirror manifolds. In particular we obtain a mathematical
framework which explains the construction of mirrors of rigid Calabi-Yau
manifolds.
