Similar to the absolute value or modulus symbol |x|, I need a new symbol to express division of two integers regardless of order such that numerator >= denominator to produce an answer with an absolute quotient |q| > 0 and an absolute remainder |r| >= 0.
Is there a standard symbol for such directionless division, or should the math community invent a new the symbol like a 90° rotated ÷ or |÷|?
Here is a sample usage:
What you are after is not a special division but a function,
$$\langle x\rangle=\begin{cases}|x|\le1\to\dfrac1x,\\|x|\ge1\to x.\end{cases}$$
In fact, it can be related to the absolute value by
$$\exp(|\log(x)|)$$ (the sign of $x$ being kept apart) and is indeed a useful operation. But I have no notation to suggest.