Can ANOVA be used with a categorical outcome and continuous predictor? Of course ANOVA can be used with a continuous y and a categorical x.
If on the other hand my y is categorical and my x is continuous, is using ANOVA still valid? Or is multinomial regression my only option?
My (superficial) understanding would tell me it is, as ANOVA is comparing a continuous variable acoss groups defined by a categorical variable, but I am not sure. I seem to understand from this answer to a similar question (What is the best way to test for a relationship between a binary variable and a continuous variable?) that you can't - but I don't understand the rationale.

It depends on the experimental design and what hypothesis you want to test.
ANOVA would evaluate whether the mean value of x differs among the groups identified by y.
Multinomial regression would evaluate the probability of being in a particular group in y, given a particular value of x.
Those are different hypotheses. Which do you want to test?
Yes, if knowing the value of x helps you predict the group in y, there would tend to be differences in the mean values of x among the groups. But you can't count on a one-to-one correspondence between those two types of models, particularly if there are other important covariates to consider.
If you manipulated values of x and observed the outcome classes y, then the multinomial model would be the way to go. Then you have a specific "predictor" x and a defined "outcome" y; you would want the analysis method to reflect the experimental design.
In some types of observational study, however, the distinction between the "outcome" and the "predictor" isn't always so clear. For example, you might have several types of disease and biomarkers whose values differ depending on the type of disease. In an early-stage study you might perform ANOVA on biomarker levels among the types of disease, even if you think that fundamentally the type of disease is the "outcome" of the biomarker. That's particularly the case if your candidate biomarkers are expression levels of thousands of gene products. Then a model of genes as a function of disease type might be a good way to begin.
The answer to your question thus depends on the nature of your study.
