  The canonical form of Matrix Product States (MPS) and the associated
fundamental theorem, which relates different MPS representations of a state,
are the theoretical framework underlying many of the analytical results derived
through MPS, such as the classification of symmetry-protected phases in one
dimension. Yet, the canonical form is only defined for MPS without non-trivial
periods, and thus cannot fully capture paradigmatic states such as the
antiferromagnet. Here, we introduce a new standard form for MPS, the
irreducible form, which is defined for arbitrary MPS, including periodic
states, and show that any tensor can be transformed into a tensor in
irreducible form describing the same MPS. We then prove a fundamental theorem
for MPS in irreducible form: If two tensors in irreducible form give rise to
the same MPS, then they must be related by a similarity transform, together
with a matrix of phases. We provide two applications of this result: an
equivalence between the refinement properties of a state and the divisibility
properties of its transfer matrix, and a more general characterisation of
tensors that give rise to matrix product states with symmetries.
