Rosa - bilateral integration (coordination of both sides of the body during activities) would be assessed by an OT and/or physio when assessing for developmental coordination disorder (dyspraxia).
A lot of schools have staff who can recognise dyspraxia, but they do not do the in-depth assessment that therapists can. If its spotted early and activities done in school, often the problem can be headed off early. In an ideal world we would have OTs in all primary schools to advise on this, but of course that doesnt happen.
A lot of myths are growing up about dyspraxia - eg if a child has ASD, behaviour problems, learning difficulties and also presents with dypsraxic tendencies, then a bit of OT or physio is going to sort it out because the behaviours were associated with the dyspraxia. I dont think that is necessarily true, as its often the other way round - if a child has developmental delay, learning difficulties etc, they are bound to have some elements of dyspraxia., as it means the abilty to plan movement.
Also some schools are coming to believe that if a child functions well in some areas, then they cant have dyspraxia, ASD, etc. It sounds like this is how your poor DD fell through the net - she was good at spelling and maths, so they didnt think she could be dyspraxic. TBH, if someone described to me what you have, I would have suggested getting an assessment, going via GP rather than through education. (Im a physio by the way.
Some children respond very well to treatment, but others with more complex diagnoses (eg those overlapping with language or communication disorders, ASD, ADHD) may not. But its always worth getting more information, even if therapy isnt going to be the answer.
Try googling the Dyspraxia Foundation for further information (sorry, Im no good at links!) They have some good leaflets for parents and schools.