Do reverse-thread eye screws exist?
Using a left hand threaded eye screw on one end of a cable and a right handed screw eye on the other end is an OK theoretical solution. However in practice it is nearly impractical because to avoid twisting the cable you have to tighten one may be a quarter turn and then run to the other end and turn it the opposite direction quarter turn.

If you want to avoid turn buckles then you best realistic solution is to just use a regular screw eye to secure the cable on one end. Then use a threaded eye bolt on the other end that has a long threaded end that passes completely through the post. Afix the cable to the eye and then simply tighten using a using a regular nut and washer in the threaded end of the eye bolt. This way you can get a consistent look of the cable terminations but have proper provision for re-tightening the cable at a later date.