  Mixing by cutting-and-shuffling can be mathematically described by the
dynamics of piecewise isometries (PWIs), higher dimensional analogs of
one-dimensional interval exchange transformations. In a two-dimensional domain
under a PWI, the exceptional set, $\bar{E}$, which is created by the
accumulation of cutting lines (the union of all iterates of cutting lines and
all points that pass arbitrarily close to a cutting line), defines where mixing
is possible but not guaranteed. There is structure within $\bar{E}$ that
directly influences the mixing potential of the PWI. Here we provide new
computational and analytical formalisms for examining this structure by way of
measuring the density and connectivity of $\varepsilon$-fattened cutting lines
that form an approximation of $\bar{E}$. For the example of a PWI on a
hemispherical shell studied here, this approach reveals the subtle mixing
behaviors and barriers to mixing formed by invariant ergodic subsets (confined
orbits) within the fractal structure of the exceptional set. Some PWIs on the
shell have provably non-ergodic exceptional sets, which prevent mixing, while
others have potentially ergodic exceptional sets where mixing is possible since
ergodic exceptional sets have uniform cutting line density. For these latter
exceptional sets, we show the connectivity of orbits in the PWI map through
direct examination of orbit position and shape and through a two-dimensional
return plot to explain the necessity of orbit connectivity for mixing.
