  A fourientation of a graph is a choice for each edge of the graph whether to
orient that edge in either direction, leave it unoriented, or biorient it.
Fixing a total order on the edges and a reference orientation of the graph, we
investigate properties of cuts and cycles in fourientations which give
trivariate generating functions that are generalized Tutte polynomial
evaluations of the form \[(k+m)^{n-1}(k+l)^gT\left(\frac{\alpha k + \beta l +
m}{k+m},\frac{\gamma k + l + \delta m}{k+l}\right)\] for $\alpha,\gamma \in
\{0,1,2\}$ and $\beta, \delta \in \{0,1\}$. We introduce an intersection
lattice of 64 cut-cycle fourientation classes enumerated by generalized Tutte
polynomial evaluations of this form. We prove these enumerations using a single
deletion-contraction argument and classify axiomatically the set of
fourientation classes to which our deletion-contraction argument applies. This
work unifies and extends earlier results for fourientations due to Gessel and
Sagan, and results for partial orientations due to the first author, and the
second author and David Perkinson, as well as results for total orientations
due to many authors. We conclude by describing how these classes of
fourientations relate to geometric, combinatorial, and algebraic objects
including bigraphical arrangements, cycle-cocycle reversal systems, graphic
Lawrence ideals, Riemann-Roch theory for graphs, zonotopal algebras, and the
reliability polynomial.
