Why do the splittings in mirrored COSY crosspeaks differ?
Let's focus for now on the two circled crosspeaks. In a COSY sequence, the frequency in the indirect dimension (the vertical axis) corresponds to the spin which the magnetisation is on during the $t_1$ period. The frequency in the direct dimension (horizontal axis) corresponds to the spin which you measure the magnetisation of during $t_2$.

It turns out that these crosspeaks are from butanal, so let's also colour-code the protons in butanal as follows: we have $\ce{C\color{red}{H}_3-C\color{blue}{H}_2-C\color{green}{H}_2-CHO}$.

The red crosspeak occurs at ~1.65 ppm in the indirect dimension, and ~0.95 ppm in the direct dimension. This means that the magnetisation during $t_1$ was on the blue $\ce{C\color{blue}{H}_2}$ proton, and was subsequently transferred to the red 
$\ce{C\color{red}{H}_3}$ proton during acquisition (in $t_2$). The coupling between these two sets of protons is referred to as an active coupling; it is the coupling that is responsible for generating the crosspeak.

Conversely, the the blue crosspeak arises due to transfer from $\ce{C\color{red}{H}_3}$ (during $t_1$) to $\ce{C\color{blue}{H}_2}$ (during $t_2$). So, it also has an active coupling between the same nuclei. The difference here is that the final $\ce{C\color{blue}{H}_2}$ protons are also coupled to the green $\ce{C\color{green}{H}_2}$ protons! This is what is called a passive coupling: one that does not generate the crosspeak, but is still reflected in the multiplet structure as a splitting.

The differences in the crosspeak structure arise because of differences in passive couplings to other nuclei.

Exactly how this is manifested is not easy to explain in this case, because there are multiple equivalent nuclei. But if you are keen, read on. (I'm also not going to present any quantum mechanical analysis, because that would really take way too long.)



In fact, the red crosspeak also has one passive coupling – the $\ce{C\color{red}{H}_3}$ group is coupled to not just one, but two protons adjacent to it, and the crosspeak resembles a case where one of the two protons is an active coupling partner and the other is passive.

We first deal with passive couplings. These lead to in-phase splittings, i.e. peaks which are both positive and are separated by the coupling constant, approximately 7 Hz in our case. For simplicity's sake we will stick to just the direct dimension for now. Because we only have one effective passive coupling, this leads to a doublet with a peak separation of ~7 Hz.

(The indirect dimension, broadly speaking, has similar behaviour in that it is also affected by passive and active couplings, but the exact source of these couplings can differ. Also, it is sometimes harder to observe multiplet structure in the indirect dimension because of different resolutions in the two dimensions.)

Active couplings, on the other hand, lead to antiphase splitting, where peaks have opposite signs. So, our doublet is further split into two doublets: one positive and one negative, and these two doublets themselves are 7 Hz apart. We find that the middle peak exactly cancels itself out, and we are left with one positive and one negative peak separated by exactly twice the coupling constant. This is what is observed in the red crosspeak.



On the other hand, the blue $\ce{C\color{red}{H}_2}$ protons have one active coupling (to one of the $\ce{C\color{red}{H}_3}$ protons), but the other two $\ce{C\color{red}{H}_3}$ protons are effectively passive couplings. On top of that, it has two further passive couplings to the $\ce{C\color{green}{H}_2}$ protons.

The four passive couplings lead to the formation of a $1:4:6:4:1$ quintet (we're going to assume that all the couplings have similar size, which is quite reasonable). The single active coupling causes an antiphase splitting of this quintet, which is easier to express mathematically:

1   4   6   4   1
   -1  -4  -6  -4  -1
-----------------------
1   3   2  -2  -3  -1


(imagine that there is 7 Hz separation between every column of numbers). This $1:3:2:-2:-3:-1$ pattern exactly matches how the blue crosspeak looks like.

For more information, see e.g. Claridge, High-Resolution NMR Techniques in Organic Chemistry (3rd ed.), Section 6.1.5.2.