We present an experimental and theoretical study of the polarization and time-resolved nonlinear dynamics of a long-wavelength single-mode linearly polarized vertical-cavity surface-emitting laser (VCSEL) subject to orthogonal optical injection. Special attention is paid to the correlation properties of both linear polarizations emitted by the VCSEL. We measure simultaneous time traces of both linearly polarized output signals for several values of the bias current, the strength of the optical injection, and the frequency detuning between master and the free-running VCSEL. Different dynamical regimes, including periodic, period doubling, and irregular dynamics, are observed for both polarizations. For positive frequency detuning we usually find that the orthogonal polarization is the only polarization that contributes to the dynamics of the total power. Only for large values of the bias current and for small values of the optical injection strength, near the border of the periodic region, have we found correlated periodic dynamics in both linear polarizations. For negative frequency detuning we find anticorrelated dynamics in both linear polarizations. The average and dispersion of the time between consecutive pulses that appear in the anticorrelated regime increase with the injected power. The irregular dynamics, characterized by broadened power spectra, is related to large values of the dispersion of the time between pulses. A good overall qualitative agreement is found between our theoretical and experimental results.