  Pontrjagin's seminal topological classification of two-band Hamiltonians in
three momentum dimensions is hereby enriched with the inclusion of a
crystallographic rotational symmetry. The enrichment is attributed to a new
topological invariant which quantifies a $2\pi$-quantized change in the
Berry-Zak phase between a pair of rotation-invariant lines in the bulk,
three-dimensional Brillouin zone; because this change is reversed on the
complementary section of the Brillouin zone, we refer to this new invariant as
a returning Thouless pump (RTP). We find that the RTP is associated to
anomalous values for the angular momentum of surface states, which guarantees
metallic in-gap states for open boundary condition with sharply terminated
hoppings; more generally for arbitrarily terminated hoppings, surface states
are characterized by Berry-Zak phases that are quantized to a rational multiple
of $2\pi$. The RTP adds to the family of topological invariants (the Hopf and
Chern numbers) that are known to classify two-band Hamiltonians in Wigner-Dyson
symmetry class A. Of these, the RTP and Hopf invariants are delicate, meaning
that they can be trivialized by adding a particular trivial band to either the
valence or the conduction subspace. Not all trivial band additions will nullify
the RTP invariant, which allows its generalization beyond two-band Hamiltonians
to arbitrarily many bands; such generalization is a hallmark of
symmetry-protected delicate topology.
