Greetings and welcome to the community forum!
I think it should be quite easy, although probably not obvious. OO Writer manages all page layout through page styles. The default page style--named "Default"--is configured to apply to both right (fronts, odd #s) and left (backs, even #s) pages but you can change that to right-only and it will only apply to the front sides. THen you can create a new page style for the back sides and anchor your constant text to a header or footer of that page style.
Once that's done, you can configure each page style to switch to the other, thus alternating pages.
The only part I'm not sure of is how to prevent Writer from placing any of the running text on the back sides. One way might be to put the constant text in a frame that fills the page, leaving no room for any regular text. I'd have to play with it a bit to see if there's a better/easier way.
Hmm ... it seems the hard part is getting Writer to NOT number the page backs; I assume you don't want them numbered. I think to get what you want you're going to have to print two passes, front sides and then back sides. Otherwise Writer is going to insist on including the page backs in the numbering.