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Contradictions are "impossible" in combinatorial logic. In combinatorial logic, the output of a logical formula is determined entirely by the inputs. For example

True = (false) OR (true)

In such a formula, the output of a "contradiction" is always false, regardless of the input. The classical example of a contradiction is

(A) and (not A)

Which always outputs false, in combinatorial logic. Combinatorial logic is not the most advanced form of logic available in maths. It is infact the most basic with the least computing power, and virtually every computation ever performed is at the bare minimum a sequential logic.

In sequential logic, we introduce memory, or time. In sequential logic, contradictions are not only "possible", but extensively used. The most famous such contradiction is

0 = 1

Which in sequential logic produces a square wave oscillation. Basically, if you were to create an electrical circuit to compute 0=1, the circuit would oscillate between 1 and 0, thereby creating an oscillator. This is infact how digital clocks are created!

This is very similar to how contradictions are used in dialectics. They act as "motors" which produce a time or progression effect. Basically, a society without contradictions will have no movement. In human societies, class contradictions are the primary contradictions which dominate the progression of the society.

You should note that Marx and Engels were not aware of sequential logic, so they would have explained things differently.