The paper describes the design of eight new rotationally invariant rate-2/3 and rate-3/4 convolutional codes. The rate-2/3 codes are suitable for 8PSK (8-level PSK) coded signals and are transparent to phase rotations in the received signal carrier phase, that are multiples of π/2 radians, whereas the rate-3/4 codes are suitable for 16PSK (16-level PSK) coded signals and are transparent to phase rotations that are multiples of π/4 radians. The new codes are bandwidth-efficient and, when combined with a simple carrier-phase correction system described in the paper, they are able to tolerate the likely carrier-phase changes in the reference carrier of the coherent demodulation process. Thus they avoid the prolonged error bursts that are otherwise caused in the detected data symbols by such phase shifts. Among these codes, two rotationally invariant rate-3/4 codes have been found with minimum free distances equal to those of the optimum but not rotationally invariant codes, having the same rates and constraint lengths. The majority of the codes designed here have asymptotic coding gains which are only up to about 0.5 dB inferior to those of the corresponding optimum but not rotationally invariant codes. Extensive computer simulation tests have been carried out on the various codes studied, and these have confirmed the optimum or near-optimum performances that have been theoretically predicted for these codes.