Are $((0,1]\cap\mathbb Q)\times\mathbb Q$ and $\mathbb Q \times([0,1)\cap\mathbb Q)$ order isomorphic? The ordering here isn't specified but I assume it's lexicographic.
I think the answer is yes. My reasoning is the following:
Neither sets have a least or a greatest element  because the second component can always be increased or decreased indefinitely. 
Both sets are countable as a product of two countable sets. Both sets are dense (it might be a bit tricky to prove this, but it seems relatively simple).
Therefore, both sets are order isomorphic to $\mathbb Q$, so they are order isomorphic to each other.
Does this make sense?

If the intended ordering is lexicographic, your argument is correct. In a general order-theoretic context, however, there is at least one other reasonable possibility: the question might be about the product partial order in which $\langle p_0,q_0\rangle\le\langle p_1,q_1\rangle$ iff $p_0\le p_1$ and $q_0\le q_1$. In that case they are not order-isomorphic: if $\uparrow\!\!\langle p,q\rangle=\{\langle r,s\rangle:p\le r\text{ and }q\le s\}$, then in the first set $\uparrow\!\!\langle 1,q\rangle$ is linearly ordered for every $q\in\Bbb Q$, while in the second set no $\uparrow\!\!\langle p,q\rangle$ is linearly ordered.
