Bilinear pseudodifferential operators with symbols in the bilinear analog of
all the H\"ormander classes are considered and the possibility of a symbolic
calculus for the transposes of the operators in such classes is investigated.
Precise results about which classes are closed under transposition and can be
characterized in terms of asymptotic expansions are presented. This work
extends the results for more limited classes studied before in the literature
and, hence, allows the use of the symbolic calculus (when it exists) as an
alternative way to recover the boundedness on products of Lebesgue spaces for
the classes that yield operators with bilinear Calder\'on-Zygmund kernels. Some
boundedness properties for other classes with estimates in the form of Leibniz'
rule are presented as well.