  We show that tori in Engel 4-manifolds behave analogously to knots in contact
3-manifolds: Every torus with trivial normal bundle is isotopic to infinitely
many distinct transverse tori, distinguished locally (and globally in the
nullhomologous case) by their formal invariants. (Few examples of transverse
tori were previously known.) We classify the formal invariants, which are
richer than for transverse knots, and show that these can all be realized by
any torus in an overtwisted Engel manifold, up to homotopy through Engel
structures. Fixing Engel structures not known to be overtwisted, we explore the
range of the primary invariants of given tori. A sample application is that
many Engel manifolds admit infinitely many transverse homotopy classes of
unknotted transverse tori such that each class contains infinitely many
transverse isotopy classes.
