  Tensors are multiway arrays of data, and transverse operators are the
operators that change the frame of reference. We develop the spectral theory of
transverse tensor operators and apply it to problems closely related to
classifying quantum states of matter, isomorphism in algebra, clustering in
data, and the design of high performance tensor type-systems. We prove the
existence and uniqueness of the optimally-compressed tensor product spaces over
algebras, called \emph{densors}. This gives structural insights for tensors and
improves how we recognize tensors in arbitrary reference frames. Using work of
Eisenbud--Sturmfels on binomial ideals, we classify the maximal groups and
categories of transverse operators, leading us to general tensor data types and
categorical tensor decompositions, amenable to theorems like Jordan--H\"older
and Krull--Schmidt. All categorical tensor substructure is detected by
transverse operators whose spectra contain a Stanley--Reisner ideal, which can
be analyzed with combinatorial and geometrical tools via their simplicial
complexes. Underpinning this is a ternary Galois correspondence between tensor
spaces, multivariable polynomial ideals, and transverse operators. This
correspondence can be computed in polynomial time. We give an implementation in
the computer algebra system \textsf{Magma}.
