Under the Continuum Hypothesis, every autohomeomorphism of N* lifts to M* = (N × [0,1])*, and this yields an order-reversing autohomeomorphism of H*.
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$\mathbb{M}^*$, $\mathbb{N}^*$, and $\mathbb{H}^*$
Under the Continuum Hypothesis, every autohomeomorphism of N* lifts to M* = (N × [0,1])*, and this yields an order-reversing autohomeomorphism of H*.