I've spent the last hour trying to fix one graph. I have two columns of data points, but Calc insists on graphing x vs. y, where I want y vs. x. I do not want to switch the columns physically in the table, but rather, I want to switch the association of the data sets with the axes. How do I switch this?
(So I have one column of absorbances and another of times. Right now, it has time as the y-axis, where it clearly should be the x-axis. How do I change that?)

Of course, I can physically arrange the data table so that the x-axis would be to the left of the y-axis, but that would be ridiculous as the only option. Can someone show me a way around that?

I've not found a way to change the chart once it's been created. The only way I know to do what you want is in the creation phase. When going through the dialogues, selecting chart type, etc., watch for two radio buttons marked 'Chart in Rows Columns'. Select one and get an instant preview in the left side of that dialogue. If it doesn't show what you want, select the other. Then procede with chart creation, axis labeling, etc.
Easy way to fix the one you have, rather than make a new one, is do as you yourself suggested. Reorder the columns.