  We study the long-time behavior of scale-invariant solutions of the 2d Euler
equation satisfying a discrete symmetry. We show that all scale-invariant
solutions with bounded variation on $\mathbb{S}^1$ relax to states that are
piece-wise constant with finitely many jumps.
  All continuous scale-invariant solutions become singular and homogenize in
infinite time. On $\mathbb{R}^2$, this corresponds to generic infinite-time
spiral and cusp formation. The main tool in our analysis is the discovery of a
monotone quantity that measures the number of particles that are moving away
from the origin.
  This monotonicity also applies locally to solutions of the 2d Euler equation
that are $m$-fold symmetric ($m\geq 4$) and have radial limits at the point of
symmetry.
  Our results are also applicable to the Euler equation on a large class of
surfaces of revolution (like $\mathbb{S}^2$ and $\mathbb{T}^2$). Our analysis
then gives generic spiraling of trajectories and infinite-time loss of
regularity for globally smooth solutions on any such smooth surface, under a
discrete symmetry.
