The continuity announcer promised "an intimate portrait of an inti-mate portrait being made". It was an interesting thought, because we know the Queen does not do intimacy, at least not with her subjects. Rolf Harris, on the other hand, does nothing but. The Queen by Rolf (8pm, New Year's Day), in which Her Majesty consented not only to having her portrait painted by Rolf Harris but to having her two sittings with him filmed by the BBC, could hardly fail but be a clash of sensibilities.

In a way it was also a perfect pairing. The Queen is assumed to be nine-tenths philistine in her cultural tastes; Harris, a poll showed a few years ago, is our philistine nation's favourite painter. The durability of both has been achieved by strict adherence to a code of public blandness. (If the Queen at 79 is four years longer lived than Harris, Harris, whose position in the nation is not actually enshrined in the constitution, wins the prize for keeping his TV career going for more than 50 years.) Nevertheless, HM and RH represent two styles of celebrity: the extrovert, which is celebrity's default mode, and the introvert, the Garbo option perfected by the Queen. What would happen when the gabby Aussie chatted up the tight-lipped monarch?

Initially, the conversation was as awkward as mine are with cab drivers, Rolf in this case being the cabby and Elizabeth R being me, lobbing single-sentence logs on to a sluggish conversational bonfire. Rolf, being so ancient, could remember sitting in Hyde Park with his accordion on the day of her coronation. The Queen re-called the day, too. "Cold and wet, as I seem to remember." Sydney could be hot, she said. It could also be wet, Rolf observed. Not as wet as Canada, she observed back. "Every time there's 40,000 people to see you, you get wet. Or they do."

The weather having been exhausted during the first sitting, the conversation turned canine in the second. Rolf owns two dogs. The Queen keeps corgis. "They chase people," she explained. "One of the sentries used to have a bad time." And here, to perhaps his own surprise, Harris did an impression of a dog breathlessly chasing a sentry across a parade ground.

Central to this dialogue was the understanding that the Queen does not express opinions, so when she relaxed a mite and the conversation started to not flourish but cease to wither, we viewers fell on her opinions like corgis on banquet scraps. She was not happy with the national level of animal abuse. She had read that one house kept 30 pets. Rolf topped this. On Animal Hospital they had discovered 130 cats and dogs in a flat. "Some people go on a holiday and leave their children behind," remarked the Queen, possibly humorously. Then she started her complaining. As a little girl she had found her first portraitist "horrible". As for Queen Mary, her big-haired grandmother, she had bequeathed Elizabeth the tiara for her coronation but first removed its base. "I suppose she thought," the Queen said bitterly, "'The child's only 21. She doesn't need a big one.'" Suddenly she was quite the chatterbox. As she left the room she breached her own protocol never to express a view about her portraits. She told Rolf she thought his work was "very clever" and looked "nice". A grateful BBC must have wanted to award her a Blue Peter badge.

So what did we learn about her? Mainly that she is exactly as anyone who saw Alan Bennett's A Question of Attribution thought she would be. Beneath a reserve that may have started as shyness, she has a wintry sense of humour (which is better than none at all). She is either a good liar or really is familiar with Animal Hospital. (But then we already knew from the Mirror that she is also a fan of Kirsty's Home Videos.) Her face has got so used to disguising boredom that it involuntarily also disguises interest. She gets colds and sucks cough sweets. She is no one's fool.

And Rolf? He lives in a big house by the Thames, has hair sprouting from the top of his nose, gets tired after making conversation with reigning monarchs and suffers from insomnia. After a sleepless night he got up at 5am to continue working on the portrait - having first, we are to believe, woken up a resident TV crew to film him doing so. Yet for all the artifice, Rolf, like Her Majesty, is the genuine article: in his case, obviously a thoroughly nice bloke.

His attempt in oils to turn HMQ into a jolly cobber failed, as fail it must. Her allegedly infectious smile looked, on canvas, toothsome and infectious indeed (rabies, I am thinking). But the portrait provided by the film, although hardly intimate, was a great success, managing to give us a painting lesson and a history of royal portraiture, as well as a further, if fractional, insight into a strangely ordinary woman whose face we see every time we lick a stamp or hand over a banknote.

Most of all, this carefully made film provided us with an indelible visual image: the Queen sitting doing nothing while being scrutinised for signs of humanity. Metaphors for royalty do not come neater than that.

Andrew Billen is a staff writer for the Times