Turning long term integrator into bipolar input introduce a concept similar to “negative frequency”, where the counter would react to decrement its counter when a negative frequency is supplied. Off course there’s no such concept in reality, but we can achieve the same function by rectifying the input before converted to frequency, and detecting the polarity to control the up-down mode of the counter.
The first op-amp is an absolute value converter, which convert bipolar signal into monopolar (positive swing only). The second op-amp is act as a polarity detector, to control the up-down mode of the counter. [System's block diagram source: Microchip Application Note]