  We perform numerical simulations of turbulent convection for infinite Prandtl
number with free-slip walls, and study the dynamics of flow reversals. We show
interesting correlations between the flow reversals and the nonlinear
interactions among the large-scale flow structures represented by the modes
$(1,1), (2,1), (3,1)$ and some others. After a flow reversal, the odd modes,
e.g. $(1,1), (3,1)$, switch sign, but the even modes, e.g. $(2,2)$, retain
their sign. The mixed modes $(1,2)$ and $(2,1)$ fluctuate around zero. Using
the properties of the modes and their interactions, we show that they form a
Klein four-group $Z_2 \times Z_2$. We also show that for the free-slip boundary
condition, the corner rolls and vortex reconnection are absent during a flow
reversal, in contrast to active role played by them in flow reversals for the
no-slip boundary condition. We argue that the flow reversals with the no-slip
and free-slip boundary conditions are different because they are induced by
nonlinearities $ ({\bf u} \cdot \nabla) {\bf u}$ and $({\bf u} \cdot \nabla)
\theta$ respectively.
