Can you quarter the processing time for real, symmetric FFTs?
I'm not sure about quartering the process time but this is what I can think of for now. N point DFT of x.

$X_k = \sum_{n=0}^{N-1}x_nW_N^{kn}$

By the even symmetry assumption

$X_k = \sum_{n=0}^{N/2-1}x_n(W_N^{kn}+W_N^{k(N-1-n)})$

Then we simplify

$X_k = W_N^{k(N-1)/2}\sum_{n=0}^{N/2-1}x_n(W_N^{kn}W_N^{-k(N-1)/2}+W_N^{-kn}W_N^{k(N-1)/2})$

$X_k = 2W_N^{k(N-1)/2}\sum_{n=0}^{N/2-1}x_n\cos(2\pi k\frac{n-(N-1)/2}{N})$

There are probably some errors but if simplifications are valid the 'FFT' size is reduced by a factor of 2 and the twiddle factors are real (this maybe corresponds to your quartering). As in normal FFT the symmetry and periodicity can be exploited to decompose the 'symmetric FFT' into radix-2 steps for instnace.