  We consider spherical quadrangulations -- spherical embeddings of
multigraphs, possibly with loops, so that every face has boundary walk of
length 4 -- in which all vertices have degree 3 or 4. Interpreting each degree
4 vertex as a crossing, these embeddings can also be thought of as transversal
immersions of cubic graphs which we refer to as the {\it extracted graphs}. We
also consider quadrangulations of the disk in which interior vertices have
degree 3 or 4 and boundary vertices have degree 2 or 3. First, we classify all
such quadrangulations of the disk. Then, we provide four methods for
constructing spherical quadrangulations, two of which use quadrangulations of
the disk as input. Two of these methods provide one-parameter families of
quadrangulations, for which we prove that the sequence of isomorphism types of
extracted graphs is periodic. We close with a description of computer
computations which yielded spherical quadrangulations for all but three cubic
multigraphs on eight vertices.
