Is $X\times Y$ with lexicographical ordering order-isomorphic with $X\times Y$ with anti-lexicographical ordering?

Question

I am trying to figure out if $X\times Y$ with lexicographical ordering order-isomorphic with $X\times Y$ with anti-lexicographical ordering? Is it true?

Answer 1

They're not, use $X =\mathbb{N}$ and $Y = \{0,1\}$ to see the differences (both in their usual order).