  We study the dimer and Ising models on a finite planar weighted graph with
periodic-antiperiodic boundary conditions, i.e. a graph $\Gamma$ in the Klein
bottle $K$. Let $\Gamma_{mn}$ denote the graph obtained by pasting $m$ rows and
$n$ columns of copies of $\Gamma$, which embeds in $K$ for $n$ odd and in the
torus $\mathbb{T}^2$ for $n$ even. We compute the dimer partition function
$Z_{mn}$ of $\Gamma_{mn}$ for $n$ odd, in terms of the well-known
characteristic polynomial $P$ of $\Gamma_{12}\subset\mathbb{T}^2$ together with
a new characteristic polynomial $R$ of $\Gamma\subset K$. Using this result
together with work of Kenyon, Sun and Wilson [arXiv:1310.2603], we show that in
the bipartite case, this partition function has the asymptotic expansion $\log
Z_{mn}=mn f_0/2 +\mathrm{fsc}+o(1)$, for $m, n$ tending to infinity and $m/n$
bounded below and above, where $f_0$ is the bulk free energy for
$\Gamma_{12}\subset\mathbb{T}^2$ and $\mathrm{fsc}$ an explicit finite-size
correction term. The remarkable feature of this later term is its universality:
it does not depend on the graph $\Gamma$, but only on the zeros of $P$ on the
unit torus and on an explicit (purely imaginary) shape parameter. A similar
expansion is also obtained in the non-bipartite case, assuming a conjectural
condition on the zeros of $P$. We then show that this asymptotic expansion
holds for the Ising partition function as well, with $\mathrm{fsc}$ taking a
particularly simple form: it vanishes in the subcritical regime, is equal to
$\log(2)$ in the supercritical regime, and to an explicit function of the shape
parameter at criticality. These results are in full agreement with the
conformal field theory predictions of Bl\"ote, Cardy and Nightingale.
