Are infinite products commutative?

Are infinite products commutative?

While reading a textbook, I came across the following proof (for integer
partitions into odd parts and distinct parts):
The following steps can be justified by taking finite products and then
passing to the limit:
$\begin{align} \frac{1}{1-q}\frac{1}{1-q^3}\frac{1}{1-q^5} \ldots &=
\frac{1}{1-q}\frac{1-q^2}{1-q^2}\frac{1}{1-q^3}\frac{1-q^4}{1-q^4}\frac{1}{1-q^5}\frac{1-q^6}{1-q^6}\ldots\\
&= \frac{1-q^2}{1-q}\frac{1-q^4}{1-q^2}\frac{1-q^6}{1-q^3}\ldots\\ &=
(1+q)(1+q^2)(1+q^3)\ldots \end{align}$
I'm sure that the above reasoning is fine, but I can't help but have a
problem with the equality from the first line to the second.
To me, this raises the following question(s):
Are infinite products commutative?
If not, how might the above proof be justified using limits?