  A solenoidal manifold is the inverse limit space of a tower of proper
coverings of a compact base manifold. In this work, we introduce new invariants
for solenoidal manifolds, their type and their typeset. These are collections
of equivalence classes of asymptotic Steinitz orders associated to the
monodromy Cantor action associated to the fibration of the solenoidal manifold
over its base. We show the type of a solenoidal manifold is an invariant of its
homeomorphism class. We introduce the notion of commensurable typesets, and
show that homeomorphic solenoidal manifolds have commensurable typesets. When
the base manifold in question is an $n$-torus, then there is a finite rank
subgroup of $\mathbb{Q}^n$ associated to the solenoidal manifold, and the type
and typesets for subgroups of $\mathbb{Q}^n$ are isomorphism invariants that
have been well-studied in the literature. Examples are given to illustrate the
properties of the type and typeset invariants.
