Should I Switch the Signs of a Variable Before Interaction? If I have two regressors that should theoretically have opposite signed coefficients, should I switch the sign (i.e. multiply by -1) of one of the variables before creating an interaction term?
In my work, I found the 'correct' signs for the main regressors, one positive and one negative in the non-interacted regression. When I created an interaction term between the two, the main effects kept their original signs but the interaction term was insignificant.
I switched the signs of one of the variables (multiplied each observation by -1) and re-ran the regressions. Obviously, without the interaction term, the absolute value of the coefficient on that variable remained the same but the sign had switched. However, when I added the new interaction term, the coefficient on the interaction term was now significant.
Note that these are all continuous variables and interaction terms were based on mean-centered variables as in Balli & Sorensen (2013). 
I'm not sure why this would happen. Is this acceptable? Is it normal? Should I be concerned about my data?
 A: Whenever you add a new variable in a regression analysis, all coefficients may change. They will always change unless the added variable is uncorrelated to all the previous variable. The extreme case where two or more variables are highly correlated is multicollinearity.
And about your question of switching signs of one of the variables, it will just change the signs of some of your coefficients. In fact, applying any linear transformation to any variable won't change anything; coefficients will just get transformed in the same way to keep yielding the same predictions for the same situation.
A: Resurrecting this, because previous answers discuss changing a variable by a scaling factor (eg multiplying by -1), but I think it's important to talk about the general case, changing a variable by a linear transformation (spoiler, this case changes your fit!).
If you adjust one of your independent variables by simply a scaling factor (eg x' = 2x), then as the answers above say, the fit of the model doesn't change, some coefficients will just adjust themselves. This is because the interaction term is changed by merely a scaling, which is a linear transformation. Say originally you had x and y, and their interaction i = xy. If now you say that x' = 2x, then your interaction variable is now i' = x'y = 2xy = 2i. i undergoes a linear transformation, because i' is simply twice the original i.
However, if you adjust one of your independent variables by shifting (eg x' = x + 1), then all of a sudden the interaction term can change nonlinearly, and the fit of your model can change! For example, say x' = x + 1, then your interaction variable is now i' = x'y = (x+1)y = xy + y. This is NOT a linear transformation of i, because you can't express i' in the form i' = i*A + B (where A and B are constants).
Note that shifting, just like scaling, is still an example of a linear transformation, so it's non-obvious that these two cases are going to act differently. On the surface this seems to contradict an idea you may have learned in the past, that "applying any linear transformation to a variable won't change anything". And that idea is true when dealing with linear regression without interaction terms. It's the interaction term that makes this idea no longer hold.
Places to watch out for this:

*

*Categorical or boolean variables, e.g. 0 or 1. If you just swap them, then you have a shift.

*Another example is in the OP's follow up comment where mean-centering inadvertently added a shift (-3 turned into 2).

*Others?

