We consider a gauge symmetric version of the p-spin glass model on a complete graph. The gauge symmetry guarantees the absence of replica symmetry breaking and allows to fully use the interpolation scheme of Guerra to rigorously compute the free energy. In the case of pairwise interactions (p = 2), where we have a gauge symmetric version of the Sherrington- Kirkpatrick model, we get the free energy and magnetization for all values of external parameters. Our analysis also works for even p ≥ 4 except in a range of parameters surrounding the phase transition line, and for odd p ≥ 3 in a more restricted region. We also obtain concentration estimates for the magnetization and overlap parameter that play a crucial role in the proofs for odd p and justify the absence of replica symmetry breaking. Our initial motivation for considering this model came from problems related to communication over a noisy channel, and is briefly explained.