Let $A=(B\cap C) \cup (D\cap E)$ be a given set. I am looking for a different way to write this. I guess it is somehow possible to "reorder" this by using De Morgan's relations. Unfortunately, I am not successful. Does somebody see how one can write this differently?
As pointed out by Asaf Karagila, there is no option to rewrite your expression in a simpler or more elegant way.
This can be demonstrated graphically:
The following is a Venn diagram for your case:
And here is the Karnaugh-Veitch map:
The two intersecting minterm blocks cannot be replaced by fewer or simpler minterm blocks.
You can't use DeMorgan laws here, but you can use the distributivity of $\cap$ over $\cup$ and vice versa. We have:
$$(B\cap C)\cup(D\cap E)=((B\cap C)\cap D)\cup((B\cap C)\cap E)$$
Using associativity of $\cap$ we get:
$$(B\cap C\cap D)\cup(B\cap C\cap E)$$
It's not simpler than the original form, though. Note that you can do the same thing by distributing $D\cap E$ over the other pair, so we get the following: $$(B\cap C)\cup(D\cap E)=(B\cap C\cap D)\cup(B\cap C\cap E)=(B\cap D\cap E)\cup(C\cap D\cap E)$$