I have the standard drop handlebars on my road bike. When I'm in the lower position, my head is pointing down so my eyes are looking at the road and my front tire. Looking forward requires me to tilt my head up to an awkward angle and I often get a kink in my neck after a longish ride. Are there mirrors available which I could mount on my bike which would provide me a front facing view while still keeping my head down?
Yes, they exist, though they are expensive and meant for racing rather than for general purpose riding on regular roads. One example is the View Speed Cyclops glasses. They take a lot of getting used to but in the arcane world of time-trial racing they have their adherents.
I would be very tempted to say that your bike isn't setup correctly for you. It shouldn't cause any back/neck issues after a long ride (although you don't say how long that is). If you handlebars/stem is of the correct length/height, then you shouldn't have any issues seeing in front of you- nor pains in these areas.
To answer your question- I am unaware of any mirrors, and this seems like it would be a very dangerous way to ride your bike.
I looked for something like this and couldn't find anything, so I manufactured my own using a mirror, glass cutter, an old bike light frame with a hose clamp to attach it to the handlebars, and some construction adhesive. I used aluminum channel for trim around it and it worked quite well until someone stole my bike so now I have to make another.