Other than the direction of summoning, it just sounds like a role playing thing. The eidolon is no more compelled to do what the summoner asks than the summoner is to do what the eidolon asks.
I’m not sure how I’d handle the reversed summoning thing though. Eidolon’s weren’t built to be PCs, so there would be a lot of little changes to make.
Since summons can't summon, and unless you Rule 0 this off, your summoned Summoner will have some problems with most of his class abilities and spells...
Edit: Of course, this being the Homebrew forum, you can easily word your Reversed Summoner in ways that make it work fully.
As a concept it seems to open up a can of worms about the planes. If you're fine with that no problem I guess.
Eidolon + {summoned monsters xor spellcaster} doesn't seem inherently more broken than Spellcaster + {summoned monsters xor eidolon}. There'd be a bunch of weird things like what would happen if the summoner cast some of the eidolon specific spells - do those affect the eidolon, the summoner or both now? If you're willing to write something longer than the average archetype it seems possible though.
In a less serious game, one of my players ran a wizard with a toad familiar. But the toad was the source of all the magic; a once-great enchanter, transformed into a common toad by an evil witch.
Mechanically, it was a wizard with a familiar. But it was fairly easy to reskin it as something very strange and unique.
The Haunted Spiritualist archetype is kind of what we're talking about here, flavour-wise, with the companion having a lot more influence over the classed character than usual.