  We study several exotic systems, including the X-cube model, on a flat
three-torus with a twist in the $xy$-plane. The ground state degeneracy turns
out to be a sensitive function of various geometrical parameters. Starting from
a lattice, depending on how we take the continuum limit, we find different
values of the ground state degeneracy. Yet, there is a natural continuum limit
with a well-defined (though infinite) value of that degeneracy. We also uncover
a surprising global symmetry in $2+1$ and $3+1$ dimensional systems. It
originates from the underlying subsystem symmetry, but the way it is realized
depends on the twist. In particular, in a preferred coordinate frame, the
modular parameter of the twisted two-torus $\tau = \tau_1 + i \tau_2$ has
rational $\tau_1 = k / m$. Then, in systems based on $U(1)\times U(1)$
subsystem symmetries, such as momentum and winding symmetries or electric and
magnetic symmetries, the new symmetry is a projectively realized
$\mathbb{Z}_m\times \mathbb{Z}_m$, which leads to an $m$-fold ground state
degeneracy. In systems based on $\mathbb{Z}_N$ symmetries, like the X-cube
model, each of these two $\mathbb{Z}_m$ factors is replaced by
$\mathbb{Z}_{\gcd(N,m)}$.
