Dimensions of McMullen carpets I refer you to Lectures on fractal geometry and dynamics by Michael Hochman. In Theorem 5.16, he claims that the Hausdorff dimension of the attractor of a family of similarities satisfying the open set condition is the same as its Minkowski and similarity dimensions. Further,  he defines McMullen carpets $K_{( m, n )}$ on page 37 as examples of self-affine sets.
I believe these can also be generated a family satisfying the conditions of the above-stated theorem by dividing the unit square $[ 0, 1 ]^2$ into squares of side length $1/mn$ and choosing all the relevant sub-squares along with the choice of $U = ( 0 , 1 )^2$ as the open set for which $R_{i_1, j_1} U \cap R_{i_2, j_2} U = \phi$ for distinct pairs $( i_1, j_1 )$ and $( i_2, j_2 )$. It should imply $\dim K_{( m, n )} = \mathrm{Mdim}\,K_{( m, n )}$ in particular which he goes on to show to be not true in Propositions 5.21 and 5.22.
I am confused. Am I missing something trivial here?
 A: Let's take a close look at the example he gives on page 37, as Hochman says "arising from the parameters $m=4$, $n=1$, and $D = \{(0,0), (1,1),(2,0)\}$." The effect of this decomposition on the unit square is shown in figure (a) below. Figure (b) shows (if I understand you correctly) the effect on the unit square of your IFS of similarities that you purport leads to the same set. After one iteration, there is indeed an easy relationship between the two.

Figures (c) and (d), however, show the attractors that ultimately arise from these processes. We can see clearly that they are not the same. In particular, each square in the self-similar version on the right contains a copy of the set. On the left, that copy is spread out over the entire rectangle; the result is quite different.
Of course, Hochman goes on to show that the carpet on the left has box-counting dimension $\log(6)/\log(4)$ and Haudsorff dimension $\log(1+\sqrt{2})/\log(2)$, which is a bit smaller. Interestingly enough, the set on the right has similarity dimension $\log(6)/\log(4)$; therefore your modification leads to a self-similar set with the same box-counting dimension as the original set, though the Hausdorff dimensions differ.
As another example, you might consider the slight modification arising from the parameters $m=4$, $n=1$, and $D = \{(0,0),(1,0),(2,0)\}$. Note that all we've done is shifted the middle rectangle down (since the digit is $(1,0)$, rather than $(1,1)$) so that it touches the $x$-axis. I think it's not too hard to show that the attractor is a self-similar Cantor set in the line of dimension $\log(3)/\log(4)$, while your modified version is a self-similar set in the square of dimension $\log(6)/\log(4)$. Thus, the procedure can change the box-counting dimension as well.
I guess a major point behind these examples is that the dimension of a self-affine set can be quite sensitive to the placement of the pieces. Indeed, if we slide that middle rectangle down continuously, each resulting self-affine set has dimension at least one, since it's projection onto the $y$-axis is an interval. As soon as the bottom of that middle rectangle hits the $x$-axis, however, the dimension will be $\log(3)/\log(6)$ which is less than one. Thus, the dimension of a self-affine can be discontinuous function of the input even in the presence of the open set condition.
