I have difficulty visualizing what a joint pdf of two independent random variables might look like. The one that I can think of is a cylindrical extension of a univariate Gaussian (let's say extent from x to y). However, in this case, only X is independent of Y (in the sense that the choice of Y does not affect the distribution of X) but Y is entirely dependent on X (in the same sense).
Independence is symmetric, as you can see both from the definition $P(X\cap Y)=P(X)P(Y)$ and from the symmetric form of Bayes’ theorem, $P(A\mid B)P(B)=P(B\mid A)P(A)$.
An example of independent variables that might help to visualize the concept is afforded by two univariate Gaussians with different variances. The product of their probability density functions has an elliptically shaped bulge; along each line parallel to an axis it yields a scaled version of the marginal density for that axis.