  We consider both the defocusing and focusing cubic nonlinear Klein--Gordon
equations $$ u_{tt} - \Delta u + u \pm u^3 =0 $$ in two space dimensions for
real-valued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in
the defocusing case, solutions are global and have finite global $L^4_{t,x}$
spacetime bounds. In the focusing case, we characterize the dichotomy between
this behaviour and blowup for initial data with energy less than that of the
ground state.
  These results rely on analogous statements for the two-dimensional cubic
nonlinear Schr\"odinger equation, which are known in the defocusing case and
for spherically-symmetric initial data in the focusing case. Thus, our results
are mostly unconditional.
  It was previously shown by Nakanishi that spacetime bounds for Klein--Gordon
equations imply the same for nonlinear Schr\"odinger equations.
