Does it make sense to interact two uncorrelated independent variables in linear regression? Should two continuous variables be correlated to include their interaction? I.e. does y = x1 + x2 + x1x2 make sense if correlation between x1 and x2 is weak at -0.1, and their coefficients in the regression have opposite signs (x1 β1 close to +1 and x2 β2 close to -1)?
And if the estimated coefficient on x1x2 is then +3, does this effectively mean the total effect of x2 is actually positive (with β3 = +3 offsetting β2 = -1), thus both variables have a net positive association with y?
Lastly, if trying to represent this with scatter plots like pairs(y x1 x2 x1x2) in r, wouldn't individual and time fixed effects (panel data) make any relationship impossible to see visually?
Thanks!

Definitely!
set.seed(2022)
N <- 10
x1 <- rep(c(0, 1), N) # (0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1)
x2 <- c(rep(0, N), rep(1, N)) # (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
x12 <- x1 * x2 # interaction between x1 and x2
#
y <- x1 - x2 + 5*x12 + rnorm(length(x1))

In this example, $x_1$ and $x_2$ are independent (not just uncorrelated) and have opposite signs in the regression model. However, the interaction term is important. Compare the $R^2$ or adjusted $R^2$ your models have when you regress on just $x_1$ and $x_2$ versus regressing on those variables plus their interaction.
