The OP is correct that a frequency offset is visible as a rotation of the BPSK symbols when viewed on a complex plane. This is consistent with instantaneous frequency being defined as the time derivative of phase (a change of phase over a change in time). With that we can conceptualize positive and negative frequencies as either positive or negative rotations.
Fixed patterns will appear when the rate of rotation is commensurate with the sample rate. This is independent of $T$ which is the symbol rate, as is only dependent on the sample rate used to observe the received symbols. Under those conditions there will be a finite number of possible values where the IQ values can land at those sample instances, and thus result in a fixed pattern.
If the BPSK waveform wasn't pulse shaped (meaning the normalized values for all time without a frequency offset were either +1 or -1), then the resulting pattern would be samples on the unit circle. However given the waveform is likely pulse shaped (as most BPSK waveforms would be for spectral efficiency), then values can be anywhere on the IQ plane within the peak to peak amplitude range of the waveform, as long as the sampling rate is multiple samples per symbol.
To understand why this occurs intuitively, consider an unmodulated BPSK waveform (continuously send 1) under the condition of a positive carrier offset of 1 Hz. The IQ constellation will be a spinning phasor rotating counter-clockwise one cycle per second (phase increasing with time linearly in the positive direction), given mathematically as $e^{j\omega{_\Delta} t}$ where $\omega_{\Delta}$ is the frequency offset in radians / sec. If we observed the waveform in continuous time, the waveform would be a circle (the unit circle on the complex IQ plane). Once sampled, the waveform will then be samples on the IQ plane. If the sampling rate was commensurate with the frequency offset, meaning the two rates are related by a ratio of integer numbers, then the resulting samples will land consistently in unique locations forming a pattern as a repetition of values will occur based on the ratio used. The simplest case is if the sampling rate was twice the frequency offset, meaning two samples for every rotation, then the resulting pattern would appear like a BPSK modulated constellation (+1 and -1)!. If the sampling rate was four times the frequency offset, then we would see a repeating pattern of +1, +j, -1, -j, or they could all be rotated from those values if there was also a phase offset, but it would be a repetition of four possible values.
If we then were to add modulation but no pulse shaping, then this means the values at any given time can take on two values as each sample can be multiplied by +1 or -1 given the state of the data symbol. This would still be a fixed pattern and independent of the data rate itself. Once we add pulse shaping, then we add many more possible states, given the intermediate values as symbols transition under condition of pulse shaping depends on a history of prior values (the number of possible transitions from one symbol to the next is $2^M$ where $M$ is the impulse response duration in symbols of the cascade of the Tx and Rx pulse shaping filters and channel). This would appear as a smearing of the pattern on the IQ plane.
