  Neutron stars are expected to have a tight relation between their moment of
inertia ($I$), tidal deformability ($\lambda$, which is related to the Love
number), and rotational mass quadrupole moment ($Q$) that is nearly independent
of the unknown equation of state (EoS) of cold dense matter. These and similar
relations are often called "universal", and they have been used for various
applications including analysis of gravitational wave data. We extend these
studies using piecewise polytropic representations of dense matter, including
for so-called twin stars that have a second branch of stability at high central
densities. The second-branch relations are less tight, by a factor of $\sim 3$,
than the relations found in the first stable branch. We find that the relations
on both branches become tighter when we increase the lower limit to the maximum
mass for the EoS under consideration. We also propose new empirical relations
between $I$, $\lambda$, $Q$, and the complex frequency
$\omega=\omega_R+i\omega_I$ of the fundamental axial $w$-mode, and find that
they are comparably tight to the I-Love-Q correlations.
