Sorry for the confusion, I'm confused myself. How about an example: It's 1AM in the morning on Tuesday and the moon is shining brightly outside your window. You query the TWC data and see the moonrise is 9PM Tuesday and moonset is 4AM Wednesday.
But the moon's already UP. To know what time the moon sets NOW (it's 1 AM, remember), you actually have to know the rise-set values from the day before, Monday.
And what values do you want to see on your skin at 1 AM Tuesday? Monday's values are more correct, since the moon hasn't set yet. Once the moon HAS set, THEN the Tuesday values are correct.
It's entirely possible to calculate the rise-set times yourself. What you may find is that, if calculating the rise-set times at 1AM Tuesday, you'll see that the moon sets at 3 AM Tuesday and rises at 9 PM Tuesday. In other words, a day where it set before it rose.
Mixing solar cycles and lunar cycles gets... loony.