Consider the following orderings on $\mathbb{Z}^2$.
Say $(a, b) \leq_1 (c, d)$ if $a \leq c$ or if $a = c$ and $b \geq d$. So for instance $$(1,3) <_1 (1,2) <_1 (1,1) <_1 (2, 3) <_1 (2,2) <_1 (2,1) <_1 (3, 3) <_1 (3, 2) <_1 (3,1)$$ In other words, this could maybe be called "the lexicographic ordering on $(\mathbb{Z}_\leq) \times (\mathbb{Z}_\geq)$," though I'm hoping for something less wordy. (Of course this actually gives an ordering on $X^2$ for any ordered set $X$.)
Say $(a, b) \leq_2 (c, d)$ if $a + b < c + d$ or if $a + b = c + d$ and $a \leq c$. So for instance $$(1,1) <_2 (1,2) <_2 (2,1) <_2 (1, 3)$$ There's also the related ordering where we replace "$a \leq c$" with "$a \geq c$."
Do these have straightforward names, or commonly-used symbols? For instance the partial ordering $(a, b) \leq_3 (c, d)$ if $a \leq c$ and $b \leq d$ is known as the product or Pareto ordering.