Are Dolby/DTS "cross setups" common?

[tripletopper]

First of all let's say I do not have a traditional set surround sound machine. I use my Xbox One and the Dolby Atmos headphone app and the dtsx headphone app and put on my headphones to get surround sound through my headphones.

I don't know what most other people do but I assume that most equipment is either DTS or Dolby and usually doesn't have both converters in it.

Is it correct to assume that you want Dolby converters with Dolby encoded discs and DTS decoders with DTS encoded discs?

If that's the case then why does Xbox not have a selection called best match or automatic where if the movie is in Dolby it turns on Dolby Atmos headphone app and if the movie is in DTS it turns on DTS X headphone app?

People both Dolby Labs and the company that owns DTS both say their headphones surround sound apps work best with movies encoded with that language and they would even go so far as to say that they would actually recommend for the best sound overall to keep likes with likes as opposed to 100% using their own product.

As I said I'm not in the traditional world of consuming audio video. I'm off at my own world with the Xbox surround apps for Blu-rays. I have no idea what's going on but does what I say seen to make sense?