The Gate ability could summon the Eidolon, although the Eidolon would be the same creature as usual.
There is nothing in the Summoner rules or Eidolon rules to suggest that Eidolons have some stronger "true form" that exists elsewhere. The class features mention various ways to call/send the Eidolon from/to its home plane, implying that the Eidolon is not leaving some part of itself behind. For all functional purposes, the Eidolon is the creature itself.
While the word "aspect" can refer to a "part" of something, it can also refer to the "form" of something, i.e., how it appears or manifests.
Since there is nothing suggesting that the Eidolon has some greater part elsewhere, I think a more likely interpretation of "aspect" is a reference to how the Eidolon manifests (in terms of its characteristics and evolutions) when summoned away from its home plane.
However, the good news is that a Summoner could potentially summon their Eidolon using their Gate spell-like ability. The spell text says:
Calling Creatures: The second effect of the gate spell is to call an extraplanar creature to your aid (a calling effect). By naming a particular being or kind of being as you cast the spell, you cause the gate to open in the immediate vicinity of the desired creature and pull the subject through, willing or unwilling. Deities and unique beings are under no compulsion to come through the gate, although they may choose to do so of their own accord. This use of the spell creates a gate that remains open just long enough to transport the called creatures. This use of the spell has a material cost of 10,000 gp in rare incense and offerings. This cost is in addition to any cost that must be paid to the called creatures.
Eidolons are named outsiders from another plane, so they should qualify as extraplanar creatures. Note that this method would cost 10,000gp worth of material components.