  We analyze entanglement in the family of translationally-invariant matrix
product states (MPS). We give a criterion to determine when two states can be
transformed into each other by SLOCC transformations, a central question in
entanglement theory. We use that criterion to determine SLOCC classes, and
explicitly carry out this classification for the simplest, non-trivial MPS. We
also characterize all symmetries of MPS, both global and local (inhomogeneous).
We illustrate our results with examples of states that are relevant in
different physical contexts.
