the time reversed brownian-motion is still a brownian motion.
see for instance this (example 15.5)
Now if your time series displays mean-reversion it is not BM.
I am not sure about what exactly you mean by a mean-reverting time series to display momentum but I am pretty certain the correct answer is no in the sense you have no general answer to this type of question in actual market time series.
Certainly in the quant world people are interested by knowing if time series exhibit mean-reversion or trending tendencies.