We consider the performance of time-division duplex (TDD) massive multiple-input multiple-output (MIMO) with imperfect calibration of the transmit and receive radio frequency chains. By deriving the achievable signal-to-interference-plus-noise ratio & x00A0;(SINR) and the per-user bit error rate & x00A0;(BER) for constant modulus constellations, we establish that, under linear precoding, reciprocity imperfections can result in substantial reduction of the array gain. To mitigate this loss, we propose an algorithm for blind estimation of the effective channel gain at each user. We show that, with sufficiently many downlink data symbols, our blind channel estimation algorithm restores the array gain. In addition, the proposed blind gain estimation algorithm can improve performance compared to standard hardening-based receivers even under perfect reciprocity. Following this, we derive the BERs for non-constant modulus constellations, viz.& x00A0;-PAM and -QAM. We corroborate all our derived results using numerical simulations.