B737-800 Captain reported a fume event during pushback on to the ramp where a 787 aircraft was starting engines resulting in the injection of engine exhaust. The event was caused by the tail to tail configuration during engine start.

Narrative

We were cleared to push straight back from Gate X and then Aircraft Y; a 787; was cleared to push from Gate Y. We were pushed tail to tail with a wind of 20 knots from around 130 degrees giving the 787 a right quartering headwind. We started without knowing the position of the 787 and called for taxi. We were told to hold. The 787 started and his exhaust was injected by our engines and filled the entire aircraft with fumes. By the time we realized this; it was too late to turn off our packs. I asked ramp to taxi around to the south line to get out of the fumes but was informed that I would not be allowed until the 787 push crew was disconnected. I informed ramp of the fume event and was told that this is the way they do it; and I would have to wait. The total disregard from ramp was a big factor. We could have moved at minimal power; or the priority of push could have been established before pushing and one of us could have waited at the gate.Pushing tail to tail is not safe and should not be continued especially with a strong wind from a starting aircraft. Ramp should be able to use good judgement in these situations; and should listen to crew concerns.