It appears that the usual C/C++ "extern" syntax works in Arduino as if the sketch file were a .cpp file:
Sketch:
int device = 123;
SomeClass.cpp:
extern int device;
void SomeClass::checkTimedEvent() {
someDevice = device; // variable from sketch
// will display "123" (or current value of device)
// if serial output has been set up:
Serial.println("device: " + String(device));
}
You may need to worry about startup and initialization order, depending on the complexity of your project.