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