The short answer is that officially, vampires (even those with souls) in the Buffy-verse don't have reflections. Obviously in the real world it's not always possible for a show made on a limited budget to "paint out" every single reflection on every single surface and occasionally you'll get a production goof.
I'm struggling to find an scripted example of Spike not having a reflection but we can certainly use Angel as an analogue given that he is similarly 'ensouled':
Angel : I’m not what you think.
Rebecca: You’re not? Because – no reflection, - dark, private office, instantly knowing those letters weren’t written in blood, I guess what I would think is - vampire.
Angel: Then again...
Rebecca: Which is impossible. Bela Lugosi, Gary Oldman, they’re vampires.
Angel: Frank Langella was the only performance I believed but...
Rebecca: This is real. - You’re real. – Do you drink blood? Eternity
With regard to that specific episode, the best I can determine is that it's a simple production goof. Wood should have known that he was a vampire from his lack of reflection in the rearview mirror. In the original script, the reveal comes when Spike 'vamps out', rather than during the awkward car ride:
Wood is making his way across the room to Xander, and he looks over to see:
SPIKE YELLING with pain, and goes into VAMP-FACE.
ON WOOD (freezing as he realizes)
WOOD : He's a vampire.
That leaves two options; either he could see Spike in the mirror (which conflicts with the accepted canon) or he couldn't see Spike in the mirror but for some reason failed to notice his lack of reflection, perhaps because it was too dark to see clearly or because the mirror was dirty.