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.

The greatest problem with your system, I think, is that the distributive law fails to hold:
$$ 1 = 1\cdot 1 = (-1+2)\cdot 1 \ne (-1)\cdot1+2\cdot 1 = 1+2 = 3 $$
This is much more important than failure of your multiplication to have nice properties in itself.
If we have a weird "multiplication" operation that is at least associative and distributes over addition like it should, that would be a (non-commutative) rng, and there is good theory for those available. And there are various well-studied structures that dispense with associativity of multiplication (non-associative algebras), but structures with two binary operations that don't distribute are not mainstream, to say the least.
What you do have seems to be a left near-ring, though.
