I’m not going to take sides here as to who was right or wrong on the arrest incident, but the police unions’ criticism of President Obama’s remarks raised an interesting question.

Police nationwide are reeling from President Obama’s televised reprimand of Cambridge cops for “acting stupidly” in the arrest of Harvard Professor Henry Louis Gates Jr.

“My response is that this department is deeply pained,” said Cambridge Police Commissioner Robert Haas yesterday. “It deeply hurts the pride of this agency.”

Local union officials representing Cambridge Sgt. James Crowley, 42, would not rule out taking their outrage over Obama’s remarks to the picket line next month when the first family lands on Martha’s Vineyard for a vacation.

I seem to recall that it was only a few months ago that the Obama Administration refused to attend the U.S. Conference of Mayors in Providence, R.I. because of a local labor union dispute. Here’s what White House Press Secretary Robert Gibbs said:

“We have always respected picket lines, and administration officials will not cross this one.”

Obviously this begs the question: will the Obamas go on vacation if the police unions picket Martha’s Vineyard?