Without a sketch or something of how you've wired things up it's hard to say but I'm going to go out on a limb and say from your description your fighting polarity, vis the lights only work when the car is facing one direction but not that other. Not knowing what you've done I'm going to assume what you haven't done. To make the LEDs work regardless of which direction the car is facing along with the LEDs and dropping resistors you'll need a bridge rectifier and a small capacitor would be good too (to help with flickering). It's quite late so I'm not going to look for circuits to show you right now... However even though I said that this is essentially what you need to do:
https://www.coursehero.com/qa/attachment/3974530/The AC side (the wavy bit on the left side) is the track pickup on your car. The polarity changes changes depending on which way the car is facing. D1 to D4 is the bridge rectifier. Load is your LED's and resistors. C is a capacitor to help smooth things out (will reduce flicker). Depending on how many LED's are in the car you could make the bridge rectifier out of LEDS (they are diodes after all). I could draw that one out later if you wanted but right now it's the day after I stared typing this post and I'm out of here.
- Tim