  We present lattice calculations of the low-lying spectrum of $^{12}$C using a
simple nucleon-nucleon interaction that is independent of spin and isospin and
therefore invariant under Wigner's SU(4) symmetry. We find strong signals for
all excited states up to $\sim 15$~MeV above the ground state, and explore the
structure of each state using a large variety of $\alpha$ cluster and harmonic
oscillator trial states, projected onto given irreducible representations of
the cubic group. We are able to verify earlier findings for the $\alpha$
clustering in the Hoyle state and the second $2^+$ state of $^{12}$C. The
success of these calculations to describe the full low-lying energy spectrum
using spin-independent interactions suggest that either the spin-orbit
interactions are somewhat weak in the $^{12}$C system, or the effects of
$\alpha$ clustering are diminishing their influence. This is in agreement with
previous findings from {\it ab initio} shell model calculations.
