  We give both efficient algorithms and hardness results for reconfiguring
between two connected configurations of modules in the hexagonal grid. The
reconfiguration moves that we consider are "pivots", where a hexagonal module
rotates around a vertex shared with another module. Following prior work on
modular robots, we define two natural sets of hexagon pivoting moves of
increasing power: restricted and monkey moves. When we allow both moves, we
present the first universal reconfiguration algorithm, which transforms between
any two connected configurations using $O(n^3)$ monkey moves. This result
strongly contrasts the analogous problem for squares, where there are rigid
examples that do not have a single pivoting move preserving connectivity. On
the other hand, if we only allow restricted moves, we prove that the
reconfiguration problem becomes PSPACE-complete. Moreover, we show that, in
contrast to hexagons, the reconfiguration problem for pivoting squares is
PSPACE-complete regardless of the set of pivoting moves allowed. In the
process, we strengthen the reduction framework of Demaine et al. [FUN'18] that
we consider of independent interest.
