  We consider the $\mathbb{Z}_2$ toric code, surface code and Floquet code
defined on a non-orientable surface, which can be considered as families of
codes extending Shor's 9-qubit code. We investigate the fault-tolerant logical
gates of the $\mathbb{Z}_2$ toric code in this setup, which corresponds to
$e\leftrightarrow m$ exchanging symmetry of the underlying $\mathbb{Z}_2$ gauge
theory. We find that non-orientable geometry provides a new way the emergent
symmetry acts on the code space, and discover the new realization of the
fault-tolerant Hadamard gate of 2d $\mathbb{Z}_2$ toric code on a surface with
a single cross-cap, dubbed a non-orientable toric code. This Hadamard gate can
be realized by a constant-depth local unitary circuit modulo non-locality
caused by a cross-cap. Via folding, the non-orientable surface code can be
turned into a bilayer local quantum code, where the folded cross-cap is
equivalent to a bi-layer twist terminated on a gapped boundary and the logical
Hadamard only contains local gates with intra-layer couplings. We further
obtain the complete logical Clifford gate set for a stack of non-orientable
surface codes. We then construct the honeycomb Floquet code in the presence of
a single cross-cap, and find that the period of the sequential Pauli
measurements acts as a $HZ$ logical gate on the single logical qubit, where the
cross-cap enriches the dynamics compared with the orientable case. We find that
the dynamics of the honeycomb Floquet code is precisely described by a
condensation operator of the $\mathbb{Z}_2$ gauge theory, and illustrate the
exotic dynamics of our code in terms of a condensation operator supported at a
non-orientable surface.
