The eye lingers on three incidents.  A banner at a pro-Palestinian rally shows a picture of Christ carrying the cross, beneath which are the words: “do not let them do the same thing today again”.

A convoy of cars displaying flags of Palestine drives along the Finchley Road, and a man shouts through a tannoy from the top of one: “f**k the Jews, rape their daughters”.

And a policewoman on another pro-Palestinian rally is embraced by a member of the crowd, before raising an arm, clenching her first, and chanting: “free, free Palestine”.

The eye may linger, but it soon leaves.  Only a tiny proportion of Britain’s population is Jewish, and Israel and Palestine are a long way away.  Nonetheless, these incidents are worth a second look, because they have implications for all of us here now.

First, the banner – and the question of double standards.  Incitement to religious hatred is a public order offence.  Why do the police, say, arrest a 71 year-old Christian evangelist who was preaching in the street, but leave this banner-holder alone?

The question has answers that may convince.  The evangelist was speaking on his own, and was the subject of a complaint.  The placard was one in a mass of others, and no objection was raised as far as we know.

Furthermore, the police’s main responsibility at a march is to maintain public order.  Their instinct would be to think twice before wading into a crowd.

And would all police would have the religious literacy to understand what the placard meant?  We doubt it – even though the reference was clearly to the blood libel.

There is also a free speech dimension.  Public attitudes on the matter vary.  In the wake of a terror attack such as 7/7, there is a rush to constrain it.

Later, the ferret tends to reverse: for example, if a student’s supervisors insist that he was wrongly arrested for downloading terrorist material, which they say he was reading while studying for a degree.

These issues are not only academic, in more than one sense of the word.  Especially with the Online Harms Bill due to come soon to the Commons.

There is often a case for changing the law, and always one for not doing so. Consider the Commission for Countering Extremism proposals in its legal review.

For example, one can publish and distribute material to intentionally stir up racial or religious hatred as long as it avoids being threatening or abusive.

That leaves Islamist groups free to distribute such material denouncing “the kuffar” (i.e: non-Muslims, largely white), and fascist ones free to do the same to black people (of all faiths and none).

If our first item raises trade-offs between free speech and public order, the second is more clear-cut. The deployment of a convoy of this kind is new in Britain, and may not be an idea that its drivers dreamed up themselves.

And those concerned were not only harrassing Jewish and other people, especially women, by shouting about rape, but by deploying a convoy.

The gesture spoke as loudly as the words, and said: “we own the streets”.  The good news is that there have been arrests in the wake of the incidents.

The less good news is that this is unlikely to be the last.  So the smarter use of police intelligence will be required, precisely in order to ensure that convoys like these don’t own the streets, whatever cause they may claim to champion.

Finally, Nusheen Jan, the policewoman who chanted “free, free Palestine” may be subject to a gross misconduct charge.  But hang on a moment.  If she is to be disciplined or fired, what about the officers who “took a knee” last year, during the Black Lives Matter protests?

It is all very well arguing that the latter was not offensive.  It all depends on one’s point of view.  The most prominent item of Black Lives Matter political programme is “defunding the police”.

One would have thought that police officers would have hesitated to kneel before those who want to take away their jobs.  But whether so or not, at stake in both cases is not only what happens in public space but in public services, and not merely within the police force.

For example, the Permanent Secretary at the Ministry of Defence, Sir Stephen Lovegrove, has signed off a letter about racial inequality to staff with a Black Lives Matter hashtag – contrary to the civil service code, which requires impartiality.

That last word is a live issue at the BBC.  Last year, it issued guidance to employees, saying that they should not “express a personal opinion on matters of public policy, politics, or controversial subjects”.

Emily Maitlis has breached them at least once, tweeting this year: “If failing to quarantine properly is punishable by 10yrs in prison, what is the punishment for failing to properly protect the country from a pandemic?”

Fair enough, you may say.  Or your view may be otherwise. But either way, Maitlis’ view wasn’t the point: rather, it was expressing it, despite the BBC’s guidance.  She hasn’t been disciplined.

The Lovegrove and Maitlis incidents cast light on the reach and limits of free speech.  Context is crucial.  For example, a student in a university should be free to express his view within the law (hence the Government’s moves to protect it).

A Permanent Secretary is not remotely in the same position.  His duty to professional impartiality over-rides his right to free speech – at least, when doing his job.

If not, then what’s to stop Sir Philip Barton, the Permanent Secretary at the Foreign Office, from signing off his e-mails with a “Free Palestine” or “Stand with Israel” hashtag?  Or Maitlis from chanting “Black Lives Matter”, fist clenched, as she opens a broadcast?

There are implications for civil society as well as public services.  These extend, for example, to sport.  The question of when politics should be put into or taken out of it isn’t straightforward to answer.

We believe that Marcus Rashford, for example, is on the right side of the line and that Hamza Choudhury, to take another, is not.  Rashford does his politics outside the football ground.  And he is careful while pursuing issues not to get partisan.

Rashford sitting at meetings of a Child Poverty Task Force, a long way away from Old Trafford on a match day, is one thing.  Choudary part-holding a Palestinian flag at Wembley in the wake of a cup final is another.  As it would be were a footballer to brandish an Israeli one.

Should players “take a knee” before games?  On the one hand, football clubs aren’t state institutions.  On the other, Black Lives Matter, as we’ve seen, is about a lot more than some fans racially abusing players on social media (or players abusing each other).

Now that supporters are returning to stadiums, there’s evidence that some resent the practice.  It won’t do to write them all off as racists.  Or do the same to all those who object to museums becoming “polyphonic spaces for critical dialogue about the pasts and the futures”.

Or to Stonewall’s campaigning in football and elsewhere (who include some gay people).  Or to the National Trust taking up Black Lives Matter, too.

The idea of neutrality in our public space is as precious in theory as it is thorny in practice.  We began by claiming that there are implications for all of us in a single policewoman shouting “free, free Palestine”, and think we have proved it.