Do 'symmetric integers' have some other name? $-1 \cdot -1 = +1$, but there seems to me to be no reason we couldn't define a number system where negative number's and positive numbers were completely symmetric.
Where:
$$1 \cdot 1 = 1$$
$$-1 \cdot -1 = -1$$
I understand that in order to do this, multiplication could no longer be commutative and we'd have to decide what the result of $1 \cdot -1$ should be. I think we could choose that resulting sign of a multiplication could be the sign of the second term, so:
$$1 \cdot -1 = -1$$
$$-1 \cdot 1 = 1$$
or more generally, the sign of any multiplication is determined by the sign of the second term.
But where they otherwise behave roughly as expected, i.e. $1 - 2 = -1$.
Some other consequences I'm aware of: 
$$\sqrt{-1} = -1$$
$$\sqrt{1} = 1$$
$f(x) = x^2$ would behave in a way that can only be described piecewise in the normal reals as $x^2$ when $x \geq 0$, and $-(x^2)$ when $x < 0$.
Is there already research or another name for such a number system? Or perhaps is there a ring that matches this? After looking at the properties of a ring, on http://en.wikipedia.org/wiki/Ring_%28mathematics%29 what I've described cannot be a ring since it does not have a multiplicative identity. There is no element i_m such that a * $i_m = a$ and $i_m \cdot a = a$ since multiplying by $1$ in the system I've described may change the sign of $a$ to be positive.

Note that elements of a ring which satisfy $x^2=x$ are called idempotents, and these become important in some contexts - for example a square matrix with $1$ as the top left entry and zero everywhere else is a non-trivial idempotent. Such things become significant, for example, in representation theory.
[note $0,1$ are trivial idempotents and in characteristic $2$ you have $-1=1$ being a trivial idempotent - I'm addressing other situations]
One issue with idempotents is that if they are invertible, then you trivially have $x=1$ simply by multiplying through by the inverse. Your $-1$ is ambiguous, because it is unclear whether you imagine it to be invertible.
Algebraic structures where $ij=-ji$ for significant elements are also important. The Quaternions are a significant example. Note that (provided your system is associative) you have $i^2j=ji^2$ so some products don't have the minus sign. You can have $i,j$ invertible in this kind of system, but then they can't be idempotent.
So there are mathematical structures of significance which capture the ideas in your post, but there are come challenges to having everything you want at once, and also a big question of what use it might be if you did.
