  We study cocycle twists of a 4-dimensional Sklyanin algebra $A$ and a factor
ring $B$ which is a twisted homogeneous coordinate ring. Twisting such algebras
by the Klein four-group $G$, we show that the twists $A^{G,\mu}$ and
$B^{G,\mu}$ have very different geometric properties to their untwisted
counterparts. For example, $A^{G,\mu}$ has only 20 point modules and infinitely
many fat point modules of multiplicity 2. The ring $B^{G,\mu}$ falls under the
purview of Artin and Stafford's classification of noncommutative curves, and we
describe it using a sheaf of orders over an elliptic curve.
