  Depending on its geometry, a spherical shell may exist in one of two stable
states without the application of any external force: there are two
`self-equilibrated' states, one natural and the other inside out (or
`everted'). Though this is familiar from everyday life -- an umbrella is
remarkably stable, yet a contact lens can be easily turned inside out -- the
precise shell geometries for which bistability is possible are not known. Here,
we use experiments and finite element simulations to determine the threshold
between bistability and monostability for shells of different solid angle. We
compare these results with the prediction from shallow shell theory, showing
that, when appropriately modified, this offers a very good account of
bistability even for relatively deep shells. We then investigate the robustness
of this bistability against pointwise indentation. We find that indentation
provides a continuous route for transition between the two states for shells
whose geometry makes them close to the threshold. However, for thinner shells,
indentation leads to asymmetrical buckling before snap-through, while also
making these shells more `robust' to snap-through. Our work sheds new light on
the robustness of the `mirror buckling' symmetry of spherical shell caps.
