Exclusive production of $etapi^-$ and $etapi^-$ has been studied with a $191,textrm{GeV}/c$ $pi^-$ beam impinging on a hydrogen target at COMPASS (CERN). Partial-wave analyses reveal different odd/even angular momentum ($L$) characteristics in the inspected invariant mass range up to $3,textrm{GeV}/c^2$. A striking similarity between the two systems is observed for the $L=2,4,6$ intensities (scaled by kinematical factors) and the relative phases. The known resonances $a_2(1320)$ and $a_4(2040)$ are in line with this similarity. In contrast, a strong enhancement of $etapi^-$ over $etapi^-$ is found for the $L=1,3,5$ waves, which carry non-$qbar q$ quantum numbers. The $L=1$ intensity peaks at $1.7,textrm{GeV}/c^2$ in $etapi^-$ and at $1.4,textrm{GeV}/c^2$ in $etapi^-$, the corresponding phase motions with respect to $L=2$ are different.