 Cardinality of the family of strict total orders

For an infinite well-orderable cardinal number $\kappa$, each strict total order is a subset of ${\kappa }^2$, giving at most $2^{\kappa }$ possibilities by the square of an infinite cardinal. For a lower bound, order $\kappa$ two-element blocks by the ordinal index and independently reverse each block according to a subset of $\kappa$. Transport along $\kappa \times 2\cong \kappa$. Distinct subsets give distinct orders. The usual axiom of choice development supplies these product identities; for well-orderable cardinals they are already provable in ZF.