We study multidimensional analogues of quasi-twisted codes from different points of view. Their concatenated structure allows us to characterize self-dual and complementary-dual classes of such codes as well as to show that multidimensional quasi-twisted (QT) codes are asymptotically good, together with their self-dual and complementary-dual subclasses. They are naturally related to nD convolutional codes as well. It is known that the minimum distance of quasi-cyclic codes provides a lower bound on the free distance of convolutional codes. An analogous result was shown for certain 1-generator 2D convolutional codes by using quasi-2D-cyclic codes. We prove a similar relation between convolutional codes and the related QT codes first, and then generalize the relation further to certain product convolutional codes and the related product QT codes, which improves the previous result in terms of dimension and number of generators. We also provide two-dimensional ternary and binary codes of modest lengths which yield good parameters.