The energy spectrum of neutrino-induced upward-going muons in MACRO was analysed in terms of relativity principles violating effects, keeping standard mass-induced atmospheric neutrino oscillations as the dominant source of v(mu) -> v(tau) transitions. The data disfavor these possibilities even at a sub-dominant level; stringent 90% C.L. limits are placed on the Lorentz invariance violation parameter vertical bar Delta v vertical bar < 6 x 10(-24) at sin 2 theta(v) = 0 and vertical bar Delta v vertical bar < 2.5 divided by 5 x 10(-26) at sin 2 theta(v) = +/- 1. The limits can be re-interpreted as bounds on the Equivalence Principle violation parameters.