Make sure your document is set up with Facing Pages checked.
Thanks for the swift reply. In fact, I did select Facing Pages when creating the document. Here are the steps I went through. I created the document with (among other options) Facing Pages checked. The document by default created one page. I then set up the master pages as basically mirrored pages using the same elements. Then I added another page to the document so that I had two document pages. By default, InDesign treated these two pages as two separate spreads (right page 1 in the first spread, left page 2 in the second page). As long as they were separate, the master pages were properly reflected (left to left, right to right). When, however, I docked the two document pages into a single spread, the right master page manifested on both pages.
As I mentioned, if I were only occasionally making these two-page spreads, I would simply place the master elements on the document pages themselves. But I must prepare about a score of these two-page spreads a week, and the continual copying and pasting of the master-page elements are tedious.
Suppose you begin with a new document with 2 pages. They default to page numbering 1 as the first page. If you select the first page icon; go to the upper-right corner of the Pages panel menu button and click on "Numbering and Section Options" you can change the first page to begin numbering at 2. This will cause it to become a left-hand page, and the number will be 2; and the second page will be a right-hand page numbered 3.
IOW, do not drag the pages to dock them.
As with so much of life, your answer seems so obvious once it's obvious, but one doesn't know what one doesn't know until one knows it. Your solution works perfectly fine, and you've saved me much tedious copying and pasting. I much appreciate your taking the time to respond.
You are entirely welcome!