Do antisymmetric FIR filters preserve the shape of their input waveforms?
Such phase response is called Generalized linear phase, where you are allowed the $\pm \pi/2$ constant phase, and zero crossings (which add $\pi$ jumps in phase at one or more frequencies).

Filters with an antisymmetric impulse response do in fact have the $\pm \pi/2$ constant phase.

Simplest case: $h[n] = \delta[n] - \delta[n-1]$, with frequency response $H(e^{j\theta}) = 2j e^{-j \theta /2} \sin( \theta/2 )$.

In practical terms, these filters are types of differentiators: they all have a zero at $z=1$ (0 frequency) due to the fact that $\displaystyle\sum h[n] = 0$.

It is clear that these filters do not retain the waveform of the signal.

Normally you say that linear phase filters "retain the waveform" in case of low-pass filters, where you want to filter out noise but retain the shape of a pulse. More importantly, you want a good eye-pattern when decoding, and a non-linear-phase filter will result in asymmetric step responses and a poor eye pattern.