I have read that for FIR filters, the duration of the transient response is equal to the order of the filter. But, for linear phase FIR filters constructed with the windowing method, the transient is N/2. I know that this has to do with the filter impulse response being symmetric with respect to N/2 (or (N-1)/ 2 if N is even), but I don't fully understand why. Could someone provide an explanation to this?
You may be confusing "transient response" with "group delay".
I'm not sure what is your definition of transient response but it's important to know that for a linear-phase, N-tap, tapped-delay line (nonrecursive) FIR filter the first valid output sample is the Nth output sample. The delay line has to completely "fill up" with N signal samples before the output samples are valid.
The group delay of such a filter is N/2-1.
My above two paragraphs do NOT apply to recursive FIR digital filters.