Checking factor / covariate independence in ANCOVA One of the ANCOVA assumptions, as I read it here, says that:
Independent variables orthogonal to covariates. In traditional ANCOVA, the independents area assumed to be orthogonal to the factors. If the covariate is influenced by the categorical independents, then the control adjustment ANCOVA makes on the dependent variable prior to assessing the effects of the categorical independents will be biased since some indirect effects of the independents will be removed from the dependent. 
See also the third assumption here. 
Could you tell me how should I verify that this assumption is met, in a given model containing three factors and one covariate?
You can refer to SPSS or R ways of doing this.
Thank you very much
 A: One approach is to see if the covariate is correlated with the predictor variables. That is, if the ANCOVA is given by:
predicted ~ covariate + predictor1*predictor2*predictor3

Then first assess whether the covariate and the various predictor effects/interactions are correlated:
covariate ~ predictor1*predictor2*predictor3

If you find that the covariate is correlated with any of the predictor variables or their interaction, then you're violating the assumption you cite. If the covariate is categorical with more than 2 levels, you'll have to assess the correlations via multinomial regression.
A: EDIT As pointed out in the comments below, this is an answer to a different question, one about ensuring that the effect of the covariate is the same in all groups. 
You have to check for interactions of the covariate and the factors. So if the ANCOVA model was
lm1 <- lm(outcome ~ covariate + factor1*factor2*factor3)

then the you can add all the interactions and look at
lm2 <- lm(outcome ~ covariate*factor1*factor2*factor3)

followed by an F-test:
anova(lm1, lm2)

It might also make sense to add fewer interactions (especially if you don't have a lot of data), so that the large number of high order interactions don't eat up the power.
