Engine on = fine.
Engine off = it depends on the car. Check the owner's manual for towing instructions (like old mech said).
For the record, most Hondas specify flat-towing is ok up to 50 miles and speed below 35 mph. I use engine-off coasting in town, but engine-on for the highway. (driving the Odyssey auto. My manual civic gets multiple EOC's every mile, at any speed)
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11-mile commute: 100 mpg - - - Tank: 90.2 mpg / 1191 miles
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