The critical behavior of d-dimensional systems with n-component order parameter phi is studied at an m-axial Lifshitz point where a wave-vector instability occurs in an m-dimensional subspace R-m (m>1). Field theoretic renormalization group techniques are exploited to examine the effects of terms in the Hamiltonian that break the rotational symmetry of the Euclidean group E(m). The framework for considering general operators of second order in phi and fourth order in the derivatives partial derivative(alpha) with respect to the Cartesian coordinates x(alpha) of R-m is presented. For the specific case of systems with cubic anisotropy, the effects of having an additional term, Sigma(alpha=1)(m)(partial derivative(alpha)(2)phi)(2), are investigated in an epsilon expansion about the upper critical dimension d(*)(m)=4+m/2. Its associated crossover exponent is computed to order epsilon(2) and found to be positive, so that it is a relevant perturbation on a model isotropic in R-m.