Sure, you can calibrate it. Use
Polaris, the north star and this
magnetic declination calculator. Then go out at night with a flashlight (you may have to leave NYC to see Polaris, but its reasonably bright so give it a try). Then just look to see which of your two compasses points the correct number of degrees (west for NYC) from Polaris.
Polaris is less than a degree from the north celestial pole, so any time of night will do for this exercise. Max error will be +/- 0.7 deg. If you want to be more accurate, use this
Local Sidereal Time clock. Be sure to set your location (Longitude is important here). When your LST matches the Right Ascension coordinate of Polaris (2 hrs 31 mins) or 12 hours later (14 hrs 31 mins), it is precisely aligned with geographic north.
You can use the big dipper to find Polaris (look at the chart on the wiki link above), or just use your compass to look approximately north. Polaris will be the same number of degrees above the horizon as your Latitude. A fist held at arm's length subtends about 10 degrees. For NYC, count up about 4 fists looking due north and the brighthest star (its not super bright) near that location is Polaris.