  We prove two types of nodal results for density one subsequences of an
orthonormal basis $\{\phi_j\}$ of eigenfunctions of the Laplacian on a
negatively curved compact surface. The first type of result involves the
intersections $Z_{\phi_j} \cap H$ of the nodal set $Z_{\phi_j}$ of $\phi_j$
with a smooth curve $H$. Using recent results on quantum ergodic restriction
theorems and prior results on periods of eigenfunctions over curves, we prove
that under an asymmetry assumption on $H$, the number of intersection points
tends to infinity for a density one subsequence of the $\phi_j$. . We also
prove that the number of zeros of the normal derivative $\partial_{\nu} \phi_j$
on $H$ tends to infinity. From these results we obtain a lower bound on the
number of nodal domains of even and odd eigenfunctions on surfaces with an
isometric involution. Using (and generalizing) a geometric argument of
Ghosh-Reznikov-Sarnak, we show that the number of nodal domains of even or odd
eigenfunctions tends to infinity for a density one subsequence of
eigenfunctions.
