  Background: Excitations with mixed proton-neutron symmetry have been
previously observed in the $N=52$ isotones. Besides the well established
quadrupole mixed-symmetry states (MSS), octupole and hexadecapole MSS have been
recently proposed for the nuclei $^{92}$Zr and $^{94}$Mo.
  Purpose: The heaviest stable $N=52$ isotone $^{96}$Ru was investigated to
study the evolution of octupole and hexadecapole MSS with increasing proton
number.
  Methods: Two inelastic proton-scattering experiments on $^{96}$Ru were
performed to extract branching ratios, multipole mixing ratios, and level
lifetimes. From the combined data, absolute transition strengths were
calculated.
  Results: Strong $M1$ transitions between the lowest-lying $3^-$ and $4^+$
states were observed, providing evidence for a one-phonon mixed-symmetry
character of the $3^{(-)}_2$ and $4^+_2$ states.
  Conclusions: $sdg$-IBM-2 calculations were performed for $^{96}$Ru. The
results are in excellent agreement with the experimental data, pointing out a
one-phonon hexadecapole mixed-symmetry character of the $4^+_2$ state. The
$\big< 3^-_1||M1||3^{(-)}_2\big>$ matrix element is found to scale with the
$<2^+_{\mathrm{s}}||M1||2^+_{\mathrm{ms}}>$ matrix element.
