Regression with negative values, inverse regression, cost stickiness

I'm doing an analysis of cost stickiness with change in cost and change in revenue. Cost stickiness means that the percentage cost will decrease when revenue decreases is lower than the percentage cost will increase when revenue increases. (ex: revenue goes down 1%, cost down 0.5%, revenue goes up 1%, cost goes up 1%) which can provide insight about how companies can lower costs depending on revenue.

My question is:

1. to prove that cost stickiness exists, is it valid to compare the beta values (slope?) of two sample, split in two groups based on whether revenue increased or decreased (the independent will be revenue and dependent will be cost) by showing that the beta value of negative samples is smaller that beta value of positive samples?

2. in this case the dependent is cost and independent is revenue, if there is an expected cost for next year (planned), is it possible to predict the revenue from this expected number? (by doing something like inverse regression, or something like using $$y = ax + b$$ to find results in $$x = (y-a)/b)$$.

Sorry if my knowledge of statistics is rudimentary, please comment if question was vague, will update question

Interesting question!

1. Yes, your approach is sound: you could split the dataset at revenue change = 0, run separate regressions and then compare the confidence intervals of the slopes. There are other possibilities, though. You could consider fitting a nonlinear regression with a quadratic term. Or perhaps a piecewise regression if there's good reason to expect a sharp change at zero, instead of a smooth shift (as with a quadractic).

2. You can use the fitted regression(s) to predict the dependent (cost) if you know the independent (revenue). You could in principle predict the expected revenue, but this can lead to strong errors because you are optimising to reduce the error in Y conditional on X. This is discussed in more detail here and here. There is a method that would yield the same answer whether you were regression Y on X or X on Y: orthogonal regression. This is useful and important in some cases, but it has its own issues and limitations, which I discuss a bit in this answer.