How do I left and right justify on the same line?

It can be done with finesse using a Layout tab setting and leader.

Starting with a U.S. Letter (in this example) blank template, with L & R margins set to 1 inch each, do the following:

1. Zoom your document to 150% and from the View menu,
   1. Show Rules
   2. Show Layout
   3. Show Invisibles
2. Enter the word Left, two spaces, and the word Right.
   1. Position your cursor after the first space character
3. On the Layout Tab panel, open the Tabs section if not already open and click the + symbol to enter the tab stop measurement.
   1. For these two words, I used 6.0 in, Alignment Left, and the ---- selection for Leader. It won't give you the indicated dashed line — just a solid line. The Stops field can be double-clicked to select it and takes decimal incremental adjustments as needed to fine-tune your leader distance.
4. The actual paragraph alignment (e.g. Left, Justify) won't matter for the above solution.
5. Depending upon how you plan to use this solution, you may wish to isolate this in a margin-wide Text box so you can move it around after setting the Arrange panel's Text Wrap to None.

With Show Invisibles Off:

Tested with Pages v14 on macOS Sonoma 14.5

I tried VikingOSX's method, and it worked. When I chose the ----- leader, the line stayed as dashes and did not turn solid.

A way to avoid having to tinker with the stops measure might be to align center (under Format/Text in the menu bar), enter the two words with two spaces between, and repeat the dash in the middle until the words are as far apart as you want them. Unfortunately, this doesn't produce a line of en dashes - instead, it combines them into em dashes. Unless you alternate dash-space-dash-space, etc. Here are what these two options look like.

Of course, you can use whatever character(s) you like for the middle line.

🤷🏼‍♂️ A mystery. Anyway, here's what I got, using your method (with 2 inch stops to save space here):

I didn't see any settings or controls that would affect this.

You are welcome.

Send a bug report about that leader character behavior to the Pages product team via the Pages application menu.

What version of Pages on what operating system?

Pages 14.0 on Sonoma 14.5, on a M1 Pro 14" MBP.

Interestingly, you get a different result with the leader dashes when we are using the same version of Pages and macOS, on Apple Silicon.