Skewness transformation for one but not the other variable? I have two dependent variables, one is positively skewed (significantly), the other is negatively skewed (not significant). I can apply a log10 transformation to improve the skewness on the first one. 
Questions:


*

*Can I then statistically compare the two variables in a mixed ANOVA test (the two variables are steps of a factor "emotion" and I compare between factor "group"), given that one is transformed and the other isn't? 

*If not, what options do I have?

 A: I assume this is an independent samples comparison, and that you want to compare the expectation of the two groups the samples are drawn from.  If it is a paired sample, the distribution of the two groups separately is irrelevant, what is relevant is the distribution of the difference.
Let us write a model.
$$ \DeclareMathOperator{\E}{\mathbb{E}}
Y_{11},\dots,Y_{1n} \sim \text{iid with mean $\mu_1$, variance $\sigma^2$} \\
Y_{21},\dots,Y_{2m} \sim \text{iid with mean $\mu_2$, variance $\sigma^2$}
$$
Now, if you use some transform (logarithm, whatever, say $g$) of, say , $Y_2$, then you can bet that $\E g(Y_2) \not = \mu_2$, so that, even if 
the null hypothesis is true, that is, $\mu_1=\mu_2$, the corresponding null hypothesis after the transformation will not be true, so the results of the t-test applied after transforming only one group will be meaningless (it will test another null than what you want). That answer your point 1.
For the second point, multiple options:


*

*Find a compromise transformation for both groups

*Use the t-test with bootstrapping

*Use some nonparametric test

*Use a permutation test

*Probably other ideas.


For choosing between these ideas, we would need to know about your context.
