Commutativity of multiplication of cosets of the commutator subgroup

Take a group $H$ with a non-trivial commutator subgroup, and form the quotient group $H^{ab} = H/H'$. Now, take the cosets of the products of elements $a,b$ and $c,d$: $abH'$ and $cdH'$ in $H^{ab}$. When is it true that $(abH')(cdH') = (acbd)H'$?

Informally, when is it correct to commute multiplications of elements of $H$ when multiplying elements of $H^{ab}$?

This is always okay. The reason it is called $H^\mathrm{ab}$ is that it is the "abelianization" of $H$. The point of factoring out the commutator subgroup ($H'$, or $[H,H]$) is to obtain an abelian quotient. So you always have $abH' = baH'$. If you want to be technical, this is because $ba = abb^{-1}a^{-1}ba = ab[b,a]$.