If I am tracking an asset like a stock (for example), then if it goes down 50% one day, to return to its previous price it would have to go up 100% (rather than 50%). I.e.: 1 * (1 - 50%) * (1 + 100%) = 1
On the other hand if you track nominal changes, the symmetry exists (e.g. 5 + 5 - 5) but you lose the intuitive scale of the movement.
Is there a better mathematical way to preserve the symmetry of addition/subtraction while maintaining the benefits of multiplication/division?
I think the symmetry you are looking for is already there if you think of percentage change in terms of multiplication (and division) from the start.
Instead of calculating a $25\%$ increase in some quantity $Q$ by finding $25\%$ of $Q$ and adding it to $Q$, just multiply to compute $1.25Q$. To undo a $25\%$ increase you just divide by $1.25$, which is the same as multiplying by $1/1.25 = 0.8$.
Using this "multiply by $1 + $ percent change" technique it's easy to calculate successive increases and decreases and understand compound interest, exponential growth and half lives.