(I)FFT without Bit Reversal
If you're using the FFT and iFFT to perform fast convolution, yes, you can do it in-place and you can do without bit-reversing in the sample processing, but you will have to bit-reverse the transfer function $H[k]$.  You would use the DIF for the forward FFT and the DIT for the inverse FFT.
O&amp;S have the clearest treatment, in my opinion.  I am robbing some drawings from O&amp;S.
Note this DIF accepts the data input as normal order in and returns bit-reversed data in the output.  Note that the butterfly has the twiddle factor multiplied in the output of the butterfly.

Here is the DIT.  Note that it accepts the data input in bit-reversed order and returns normal order data in the output.  Also note that the butterfly exactly undoes the DIF butterfly.  It multiplies first at the input to the butterfly.


Now if you use the DIT as the iFFT, then you need to make sure you scale the input by $\frac1N$ (that can be done with the transfer function multiplication) and conjugate the twiddle factors $W_N^r$.  The multiplication in the frequency domain (between the FFT DIF and the iFFT DIT) must be done with the transfer function $H[k]$ pre-bitreversed.
In the 1980s, I wrote a very simple DIF FFT and DIT iFFT in C code.  Do you want it?  I might be able to find it.  Lemme know.
Actually, I posted the C code here before.