The alternative squares do exactly that, and the effects become way more obvious with distortion. Ultimately, changing phases on the same frequencies you need to build a square changes the volume due to phase cancellation and introduces some micro-changes to the sound which is what impacts the way the sound interacts with processing (distortion, compression, all that).

Also, using less frequencies to build a square overall will produce a softer and nicer square sound. Modern software really pushes it when building a proper square by adding a ton of high frequency sines to make it "textbook square" (something that wasn't done in older hardware due to limitations). That sharpness can be easily removed to get essentially a square that doesn't cut into your ears. Something to keep in mind if you feel the need to lowpass your squares later in the chain.