[Physicist]

the causal sequence of events probably lies at the core of our everyday notion of the "flow of time" from past to future.

I can think of a counterexample. Suppose I have two events that have "spacelike" separation, meaning that they occur far enough away from each other that a light ray can't travel from the first event in time to arrive before the second event. By construction, one event cannot be said to cause the other. In fact, the time ordering of the events is observer-dependent: in one inertial frame event "1" might happen before event "A", while in another event "A" occurs first. But in every frame, they do have a well-defined ordering in time, despite the lack of causality.

[PatrickHenry]

I'm not sure that's a true counter-example. As you correctly pointed out, each causal sequence is correctly ordered, within its own frame of reference. It has a "furure-ward" time direction (from cause to consequence). The information about one such causal event travels to the other frame where it's observed; and vice versa. Neither event is a cause of the other. Although an observer may see the distant events "later" than he sees the events in his own frame, or perhaps they may appear to be simultaneous, this isn't a reversal of cause and effect. The observer he still sees the distant events in their proper sequence because he will see the distant cause precede the distant consequence. Perhaps I missed your point.