Using methods from the conformal bootstrap, we study the properties of Noether currents in the critical O(n)O(n) loop model. We confirm that they do not give rise to a Kac-Moody algebra (for n≠ 2n≠2), a result expected from the underlying lack of unitarity. By studying four-point functions in detail, we fully determine the current-current OPEs, and thus obtain several structure constants with physical meaning. We find in particular that the terms :\!J\bar{J}\!::JJ‾: in the identity and adjoint channels vanish exactly, invalidating the argument made in [Nucl. Phys. B 419, 411 (1994)] that adding orientation-dependent interactions to the model should lead to continuously varying exponents in self-avoiding walks. We also determine the residue of the identity channel in the JJJJ two-point function, finding that it coincides both with the result of a transfer-matrix computation for an orientation-dependent correlation function in the lattice model, and with an earlier Coulomb gas computation of Cardy [Nucl. Phys. B 265, 409 (1986)]. This is, to our knowledge, one of the first instances where the Coulomb gas formalism and the bootstrap can be successfully compared.