I think, and correct me if I am wrong, that this language is basically the language of the following regular expression
$00(0 \cup 1)00 \cup 01(0 \cup 1)10 \cup 10(0 \cup 1)01 \cup 11(0 \cup 1)11$
Obv., any word in L is in the language of the regular expression by determining that |x|=|y|=1.
In the other direction, it seems that words that are not in the language of the regular expression cannot be partitioned in a way that satisfies the conditions of L.
Please check correctness.