Moorish Spain lasted for 800 years, and the Ottomans/Turks have had a presence in Europe since they took Constantinople in 1453. Looking at these two civilizations I'm increasingly struck by the fact that they don't seem to be included in Europe's vision of its history, its identity or its collective cultural memory (excluding the Christendom/marauding invaders dichotomy).
Did substantial exchange actually take place, or was it limited to the areas of Europe included within those societies' spheres of influence (the Iberian peninsula and the Balkans)?
There was certainly exchange between Europe and the Middle East. There's a large amount of evidence detailing contact between the Franks (proto-French peoples) and the Abbasid Caliphate. Charlemagne himself received an elephant from Haroun Al-Rashid, and extended talks about creating an Arab-Frank alliance to combat the two mutual enemies: the Ummayad pretender Caliphate in Spain and the Byzantine Empire.
Aside from that, quite a bit of knowledge transfer occured in the 13th and 14th century, when the theory of algebra was discovered and developed upon by the Italians. They used Arabic numerals and logic as their base for creating a lot of the algebra we see today. There's also a lot of Muslim coins in European archaeology sites, implying that trade and exchange existed on a large scale.