I think OP was referring to salary cuts. Stock is known to carry risk so it being "cut" isn't taboo, but salary, which is supposed to be guaranteed, is.
Sure but in the context of saving the company money to prevent layoffs they're identical. Companies don't cut salary because they have a found a better way to cut comp costs without upsetting workers as much.
They absolutely aren't identical. If you have a grant for, whatever, 1,000 shares a year, and the shares used to be worth $100, but now they're worth $50, the company is still giving you the same things it gave before (X% of the total value of the company). It will still have to go to the board and ask for more dilution to create new share pools at the same time as it did before. Its cash-on-hand situation is no better than before.