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 ab initio shell model calculations.