Are there any flights that fly east both from origin to destination and from destination to origin?
FlightAware suggests there are no non-stop flights from LA to Delhi.

Was certainly going to point out that we're pretty limited on ultra long-haul flights still, and it'd take a very special flight setup to make your scenario.  You'd need both:


Two locations about 180° longitude apart so that the distance going east vs. the distance going west is insignificant for both legs enough so the jet stream gains flying east pay off.
Two places at just the right latitudes.  They need to be at latitudes far enough south (in the NH) that it's not much shorter to just fly the great circle through the North Pole [or near if not 180° apart].
Here is a graph showing the ratio of the distances of the path crossing the Pole vs the distance of the constant-latitude path, for places 180° longitude apart [derivation in comments]... which shows it's 50% longer to fly around the 60° latitude circle than to fly over the Pole).
However, the further equatorward (south in the NH) you go, the less benefit you'd get from the jet stream as upper wind speeds are typically much slower further south, and you'd spend less of your flight time in the jet stream.  (Here's a current jet stream map).  


Keep in mind that the jet stream is only going to add 100-200 mph in general.  So that's only like 1/3 of the cruising speed of a large jet.  Of course the benefits will show up in both in fuel usage (and the additional weight benefits of carrying less fuel) and in travel time (people like quicker flights, and you can run more flights in a year and/or pay crew less).  So despite being a longer distance, if it cut enough time/energy it might just be worth it.

And, well go figure, it turns out you weren't far off at all!  Because there actually is a direct flight from San Francisco to Delhi.  And it looks like they do exactly as you suggested!



(Click images for larger versions)

These are Air India flights 173 and 174.  The official flight does make a jaunt from Delhi to Bengaluru and back once it completes the SFO-DEL leg.  But you could indeed go around the world from Delhi to San Francisco to Delhi without any other stops... in looks like around 36 hours.

Pretty cool!