We present the exact analytical solution of the explicitly time-periodic quantum Langevin equation describing the dual-beam backaction-evading measurement of a single mechanical oscillator quadrature due to V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne [Science 209, 547 (1980)] beyond the commonly used rotating-wave approximation. We show that counterrotating terms lead to extra sidebands in the optical and mechanical spectra and to a modification of the main peak. Physically, the backaction of the measurement is due to periodic coupling of the mechanical resonator to a light-field quadrature that only contains cavity-filtered shot noise. Since this fact is independent of other degrees of freedom the resonator might be coupled to, our solution can be generalized, including to dissipatively or parametrically squeezed oscillators, as well as recent two-mode backaction-evading measurements.