I’ve always found it easier to visualize the wheatstone bridge as the redrawn circuit below. Now it consists of 2 resistive voltage dividers (R1/R2 and R3/R4) across a voltage source, Vb. If R1,R2,R3,R4 are all equal then V1 and V2 have to be equal (both Vb/2) and there is no potential difference across the meter so it reads zero center.
If R1 and R2 are equal and R4 is a lower value than R3 then V1 is Vb/2 and V2 is less than V1. In this case if you view flow from a higher potential to a lower potential then the flow is from V1 to V2 (blue arrows) and the meter reads to the right. If you now reverse it to have R3 and R4 equal and R2 is a lower value than R1 then V2 is Vb/2 and V1 is less than V2. In this case the flow is from V2 to V1 (red arrows) and the meter reads to the left.
This is a very simplistic view of a wheatstone bridge and a commercial bridge will have ratio resistors and sensitivity settings for the actual meter but my basic model still works.