I have an ordered factor in my GLM which resulted in a significant linear relationship (.L) , with the quadratic (.Q) and cubic (.C) relationships non-significant. Could anyone please give me examples of publications that have used such results, and how they are used / interpreted? Or else what steps should be taken next in R to get more meaningful and interpretable results?

The way these results are used and interpreted vary significantly across fields and journals, so I can only really talk about my ~5 years experience in social psychology / media studies.

The first thing to do is to plot the data. I hope you've done that already - some simple graph plotting the response variable over the factor. If you have a lot of points and they're on top of each other, consider using ggplot's geom_jitter to let the points spread out some.

I'd expect the most common relationship to be what you said - there's a linear relationship between the ordinal and continuous. That means as the ordinal variable increases, the continous variable tends to move in the same direction across all values of the ordinal variable. If you were to plot the two variables (hopefully you have done so before you do statistical inference tests), the shape would roughly be a line.

This is in contrast to a quadratic relationship, which is part of a parabola. It could be high in the center and low on the sides, or vice versa, or be shaped flat on one side and steep on the other.

I've never seen a cubic term tested or reported in the literature I've read. It sounds like it's a library default. When I learned statistics for the behavioral sciences, we spent a lot of time talking about finding the right formula to fit, so I'm mildly surprised there's a default in this GLM library. What R library are you using?

I'm also kind of surprised you can't find a paper in which a GLM has a significant linear but non-significant quadratic and cubic parts. Is this the first time you're using a GLM in research? Or perhaps I've misunderstood what the GLM does.

Thank you for replying and so quickly. That's very helpful. Yes I have plotted the data, but the scatter is so great the relationships are not that obvious. Perhaps I'm using the wrong search terms, I'll have another hunt for papers.