  Given a $3$-manifold $M$ with multiple incompressible torus boundary
components, we develop a general definition of order-detection of tuples of
slopes on the boundary components of $M$. In parallel, we arrive at a general
definition of representation-detection of tuples of slopes, and show that these
two kinds of slope detection are equivalent -- in the sense that a tuple of
slopes on the boundary of $M$ is order-detected if and only if it is
representation-detected. We use these results, together with new "relative
gluing theorems," to show how the work of Eisenbud-Hirsch-Neumann,
Jankins-Neumann and Naimi can be used to determine tuples of
representation-detected slopes and, in turn, the behaviour of order-detected
slopes on the boundary of a knot manifold with respect to cabling. Our cabling
results improve upon work of the first author and Watson, and in particular,
this new approach shows how one can use the equivalence between
representation-detection and order-detection to derive orderability results
that parallel known behaviour of L-spaces with respect to Dehn filling.
