I would consider such usage correct.
I can't find any direct references that explicitly confirm it's correct, but I can't think of any reason why it wouldn't be allowed.
- [a,b,c] [d,e,f] respectively
"Respectively" is correct to use as long as all groups have an equal number of elements (and all groups are correctly ordered).
- [a,b] [c,d] [e,f] respectively
This is equally valid, as it doesn't seem to violate any characteristic of the correct use of "respectively".
Considering it in a different way.
Let's suppose it's not correct. Then you would have to use:
The cat and dog chased the mouse and squirrel, respectively, through the field and barn, respectively.
And I would argue that it's then correct to omit one occurrence of "respectively", as the omission of repetitive words is allowed when the omitted words are already implicitly understood from the context.
This can also lead us to consider your usage correct.