  Cremona maps defined by monomials of degree 2 are thoroughly analyzed and
classified via integer arithmetic and graph combinatorics. In particular, the
structure of the inverse map to such a monomial Cremona map is made very
explicit as is the degree of its monomial defining coordinates. As a special
case, one proves that any monomial Cremona map of degree 2 has inverse of
degree 2 if and only if it is an involution up to permutation in the source and
in the target. This statement is subsumed in a recent result of L. Pirio and F.
Russo, but the proof is entirely different and holds in all characteristics.
One unveils a close relationship binding together the normality of a monomial
ideal, monomial Cremona maps and Hilbert bases of polyhedral cones. The latter
suggests that facets of monomial Cremona theory may be NP-hard.
