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This paper considers a single-cell massive MIMO (multiple-input multiple-output) system with dual-polarized antennas at both the base station and users. We study a channel model that includes the key practical aspects that arise when utilizing dual-polarization: channel cross-polar discrimination (XPD) and cross-polar correlations (XPC) at the transmitter and receiver. We derive the achievable uplink and downlink spectral efficiencies (SE) with and without successive interference cancellation (SIC) when using the linear minimum mean squared error (MMSE), zero-forcing (ZF), and maximum ratio (MR) combining/precoding schemes. The expressions depend on the statistical properties of the MMSE channel estimator obtained for the dual-polarized channel model. Closed-form uplink and downlink SE expressions for MR combining/precoding are derived. Using these expressions, we propose power-control algorithms that maximize the uplink and downlink sum SEs under uncorrelated fading but can be used to enhance performance also with correlated fading. We compare the SEs achieved in dual-polarized and uni-polarized setups numerically and evaluate the impact of XPD and XPC conditions. The simulations reveal that dual-polarized setups achieve 40-60\% higher SEs and the gains remain also under severe XPD and XPC. Dual-polarized also systems benefit more from advanced signal processing that compensates for imperfections.