TITLE: Do 'symmetric integers' have some other name?
QUESTION [10 upvotes]: $-1 \cdot -1 = +1$, 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.

REPLY [10 votes]: Short answer: No, it doesn't have a name.
As Jack M points out, Alberto A. Martinez's book Negative Math:
How Mathematical Rules Can Be Positively Bent spends some time exploring the consequences of your altered sign rule. (Actually, Martinez takes the sign of the product to be the sign of the first term, not the second, but this is only a cosmetic difference.) Briefly, some points of discussion are:

The symmetry between positive and negative is attractive.
Imaginary numbers are no longer introduced by square roots. This is sometimes convenient and sometimes not. If one wishes to express rotations in two dimensions, one can fall back on an explicit rotation operator, which is no longer tied to a particular number.
Multiplication is still associative, but not commutative, similarly to the multiplication of quaternions.
Multiplication is distributive over addition on one side, but not the other.
Expressions such as $\sqrt{2+\sqrt{-121}}$ and $\sqrt{-1}^{\sqrt{-1}}$ have simpler values than before.
One needs extra absolute-value signs to state the Pythagorean Theorem, which is arguably appropriate anyway.
We must take care in expanding and interpreting expressions like $(a+b)^2$. The result has a different geometric meaning depending on the signs of $a$, $b$, and $a+b$.
Extraction of roots now has a unique solution, $\sqrt[x]a$.
Roots and powers are now exact inverses: $\sqrt[x]{a^x}={\sqrt[x]a}^x=a$.
Powers and roots now commute with one another: $\sqrt[x]{a^y}={\sqrt[x]a}^y$.
Powers and roots now commute with changes of sign: $-(a^x)=(-a)^x$ and $-\sqrt{a}=\sqrt{-a}$.
Square roots now distribute over multiplication: $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$.
Similar simplifications are obtained for logarithms.
Division, fractional exponents, and trigonometry are not considered. Coordinate geometry is sketched out to illustrate how polynomials change.

Martinez briefly remarks that the new operation might be called by a different name and symbol, perhaps "replication" and $\otimes$, to distinguish it from the traditional multiplication operation… but he doesn't really recommend that.