Symmetric saturation curves

When the left and right of the saturation curve are equal and opposite, you effect the positive and negative sides of the waveform, resulting in odd harmonics, approaching a square wave.¹

For triangle waves, every other harmonic has its polarity flipped, which results in cancellations. Triangle waves’ harmonics also fall off faster than square waves.¹

When the left and right sides are mirror images of one another, the result is even harmonics. This can result in rectification, when the negative end of the waveform has its polarity flipped.¹