  The compactly supported $\mathbb{A}^1$-Euler characteristic, introduced by
Hoyois and later refined by Levine and others, is an anologue in motivic
homotopy theory of the classical Euler characteristic of complex topological
manifolds. It is an invariant on the Grothendieck ring of varieties
$\mathrm{K}_0(\mathrm{Var}_k)$ taking values in the Grothendieck-Witt ring
$\mathrm{GW}(k)$ of the base field $k$. The former ring has a natural power
structure induced by symmetric powers of varieties. In a recent preprint,
Pajwani and P\'al construct a power structure on $\mathrm{GW}(k)$ and show that
the compactly supported $\mathbb{A}^1$-Euler characteristic respects these two
power structures for $0$-dimensional varieties, or equivalently \'etale
$k$-algebras. In this paper, we define the class $\mathrm{Sym}_k$ of
symmetrisable varieties to be those varieties for which the compactly supported
$\mathbb{A}^1$-Euler characteristic respects the power structures and study the
algebraic properties of $\mathrm{K}_0(\mathrm{Sym}_k)$. We show that it
includes all cellular varieties, and even linear varieties as introduced by
Totaro. Moreover, we show that it includes non-linear varieties such as
elliptic curves. As an application of our main result, we compute the compactly
supported $\mathbb{A}^1$-Euler characteristics of symmetric powers of
Grassmannians and certain del Pezzo surfaces.
