symbolic - Define an operator with the distributive property

I would like to define the ⊕ operator with distributive, associative, and commutative properties - so that Mathematica can symbolically simplify expressions I use it in. For example:

in := Simplify[a b ⊕ a c]
out := a (b ⊕ c)

I've already given the ⊕ operator the Flat and Orderless attributes, giving the operator the other two properties I would like, but I can't figure distribution out.

Specifically, what I'm trying to do is define an operator which has the symbolic properties of Plus, but which does not evaluate against numbers, e.g

in := 2⊕2
out := 2⊕2  

is a fully simplified expression.