Previous research has suggested that isolated, initially non-axisymmetric vortices in two-dimensional flows tend to become axisymmetric, in a coarse-grained sense, by purely inviscid mechanisms. That research, however, considered only vortices with broadly distributed vorticity. In this paper, it is shown that vortices with sufficiently steep edge gradients behave in a radically different way; in particular they can remain non-axisymmetric, apparently indefinitely. Such vortices, it is argued, are more typical in inviscid two-dimensional flows than the broadly distributed vortices previously considered, and hence the tendency for vortices to become axisymmetric is not generic to these flows.