In order topology, are "continuous bijection is increasing or decreasing" and "space is connected" equivalent?

Question

Let $\langle X, \prec \rangle$ be a linear order, and $\langle X, \tau \rangle$ be the induced order topology.

Are the following two statements equivalent?

1. Every continuous (in the sense of the topology) bijection is either increasing or decreasing (in the sense of the ordering).
2. The space $X$ is connected (in the sense of the topology).