Would it be accurate to say the mean of equal length subsets is always equal to the mean of the set?
It's always true!
Consider $n=mk$ observations, where you take $k$ mutually-exclusive groups of size $m$.
Label the observations in group $i$ as $x_{ij},\: j=1,2,...,m$.
The individual means are $\bar{x}_i = \frac{1}{m}\sum_{j=1}^m x_{ij}$.
The mean-of-means is
$\overline{\bar{x}_i} = \frac{1}{k}\sum_{i=1}^k \bar{x}_{i}$
$\hspace{.5cm}=\frac{1}{k}\sum_{i=1}^k (\frac{1}{m}\sum_{j=1}^m x_{ij})$
$\hspace{.5cm}=\frac{1}{km}\sum_{i=1}^k \sum_{j=1}^m x_{ij}$
$\hspace{.5cm}=\frac{1}{n}\sum_{i,j} x_{ij}$
which is just the overall mean of the data.
For the unequal-length case, they're also equal (that is, the mean of means equals the overall mean) if you do an appropriately weighted average when taking the mean of means.