I have a small dataset that is amenable to trend analysis. Only 1 IV with three levels, a balanced design. Both the linear and quadratic contrasts came back as significant.
What would be the appropriate approach to take regarding these (incompatible) results?
There's nothing incompatible about having both the linear and quadratic terms significant.
For example, the following pattern of means (combined with small variation about the means) would do so:
3
2
1
That the average of groups 1 and 3 differs from the mean for 2 is precisely how you tell that the effect differs from linearity. There's nothing incompatible about this. Unless your quadratic is close to symmetric about the middle group the linear term should be non-zero and unless the three means are essentially on a straight line the quadratic term should be non-zero.
I don't see any difficulty in interpretation of the quadratic term; it's the linear effect that's somewhat harder to interpret (though arguably still easier in the orthogonal framework than in the "raw" case).
[The likely reason that this didn't get an answer for a long time is that it's not really clear what you're trying to do when you say "What would be the appropriate approach to take" ... (approach to what?). If you clarified that, more help might be possible]
