A set $A$ is well-ordered (by an ordering $R$) if there is an $R$-minimal element in every nonempty subset of $A$. Call $A$ doubly well-ordered (by $R$) if $R$ well-orders $A$ and $R^c$ (converse) well-orders $A$.
Obviously if $A$ is finite than $A$ is doubly well-ordered by any strict order $R$. Are there any infinite sets with this property?
Unfortunately, no. Any infinite well-ordered set will have a subset of order-type $\omega$ (the order type of the natural numbers), and this subset has no greatest element (or, if you will, no least element in the reverse ordering).