  We present new traversable wormhole and non-singular black hole solutions in
pure, scale-free $R^2$ gravity. These exotic solutions require no null energy
condition violating or "exotic" matter and are supported only by the vacuum of
the theory. It is well known that $f(R)$ theories of gravity may be recast as
dual theories in the Einstein frame. The solutions we present are found when
the conformal transformation required to move to the dual frame is singular.
For quadratic $R^2$ gravity, the required conformal factor is identically zero
for spacetimes with $R=0$. Solutions in this case are argued to arise in the
strong coupling limit of General Relativity.
