  We introduce the notion of a quasi-matroidal class of ordered simplicial
complexes: an approximation to the idea of a matroid cryptomorphism in the
landscape of ordered simplicial complexes. A quasi-matroidal class contains
pure shifted simplicial complexes and ordered matroid independence complexes.
The essential property is that if a fixed simplicial complex belongs to this
class for every ordering of its vertex set, then it is a matroid independence
complex. Some examples of such classes appear implicitly in the matroid theory
literature. We introduce various such classes that highlight different apsects
of matroid theory and its similarities with the theory of shifted simplicial
complexes. For example, we lift the study of objects like the Tutte polynomial
and nbc complexes to a quasi-matroidal class that allows us to define such
objects for shifted complexes. Furthermore, some of the quasi-matroidal classes
are amenable to inductive techniques that can't be applied directly in the
context of matroid theory. As an example, we provide a suitable setting to
reformulate and extend conjecture of Stanley about $h$-vectors of matroids
which is expected to be tractable with techniques that are out of reach for
matroids alone. This new conjecture holds for pure shifted simplicial complexes
and matroids of rank up to 4.
