Except with between-seuject factors (gender, genotype, ...), the situation with within-subject (or repeated-measures) factors bears a unique characteristic: all subjects perform all the tasks. This leads to a natural consequence: correlation among tasks. If the response from a subject is strong, (s)he most likely performs well in any other tasks as well. And the greater the differences among subjects, the higher the correlation between any pair of conditions, and the greater the relative power of repeated-measures ANOVAs.

In the population covariance matrix,

         Cond1 Cond2 Cond3  Cond4             
Cond1      125   1   3.125   4.04          
Cond2      193   3  64.644  83.64       
Cond3       19.894   3   6.631   8.58
Cond4        18.550  24    0.77                        

(variances on the main diagonal do not have to equal the covariances)
the main diagonal is the list of variances within each condition, which are more or less equal, the homogeneity of variance. Those off-diagonal numbers are the covariances among various conditions, and their equality. This property, equality of variances and that of covariances, is called compound symmetry. Compound symmetry is a little more stringent version of sphericity: Compound symmetry guaranttees sphericity, but not vice versa. It is possible, but not likely, you would encounter cases in which sphericity is met while compound symmetry is violated.  Further the nice property makes compound symmetry more popular.

Under sphericity assumption, all the F tests for main effects and interactions are valid.

In a case of mixed design with both between-subject and within-subject factors, let's use an example to illustrate the issues: BXS(A), where A, a between-subject factor, is gender with 2 levels, and B, a within-subject factor, has 3 levels,

For factor A, variance of subject within any level (group) of A is the same as that of other levels of A.  F(b, c-1). In practice, ANOVA is relative robust against reasonable (?) violation of homogeneity of variance for between-subject factors. As the groups (gender, genotype, patient vs normal) are usually independent, compound symmetry, and thus sphericity, of covariance matrix is met when homogeneity of variance is met (all off-diagonal numbers, covariances, are 0).

Then test homogeneity of variance and normality for within-subject factor B: Interactions BXS are constant across groups. Fmax(b, (b-1)(c-1)). Calculate BXS interaction for each group, and then test the largest against the smallest. Then for each group we have a separate population covariance matrix, and they are assumed to be equal across groups. Sphericity assumption: the standard errors of the differences between pairs of B level means are constant.