But back in the constituencies, a subject far nearer to most people's hearts - education - could be a further threat to party unity.
And with an Education and Employment Secretary, Gillian Shephard, who is said to be fighting a rearguard action against John Major's determination to increase selection, the split appears to be from top to bottom.
In leafy Surrey, as much a Tory heartland as anywhere, support for government policy on selection is, at best, half-hearted.
The county ushered in comprehensives in the early 1970s under the supervision of the then education secretary, Margaret Thatcher. Last week, members of the county council's education committee unanimously backed further debate on a motion strongly supporting comprehensive education.
"I definitely would not want to return to a fully selective system," said Conservative education committee spokesperson Shirley Gladman. "There's no point in turning over the whole system and starting again. Some of our comprehensive schools have achieved very good results."
Conservative group leader Cecilia Gerrard dismisses talk of "a grammar school in every town" as simplistic.
The motion in Bournemouth calling for "more secondary-level selection, and the restoration of grammar schools" may have won backing from conference delegates in a traditional display of loyalty.
But if and when selection does return to schools in a big way, the feeling on the ground may differ.
"I have heard no suggestion from any quarter that there should be a return to grammar schools," says Peter Halls-Dickerson, founder chairman of the left-wing Conservative National Education Society. "There is a feeling that everything was well in the halcyon days of grammar schools, but I remember the problems. "
"A huge number of youngsters would have benefited if they had been in comprehensive schools at that time. When Mrs Thatcher moved towards comprehensive schools, it was something to be proud of."
Graham Pycock, chair of Lambeth Conservative Education Forum, detects a changing mood.
"Governors are beginning to have their doubts. There is the beginning of a serious groundswell, which will grow. The Government pretends everyone can get something out of going for more selection, but it's not true. More and more governors are saying they didn't realise it would mean creating secondary modern schools."
This view is by no means universal. In Dartford, Kent, which never went comprehensive, local MP and former schools minister Bob Dunn says: "It's a vote-winner in Kent and across the country.
"Parents recognise that the comprehensive experiment has failed. You've only to ask Harriet Harman or Tony Blair. People want more choice, and that's what selective schools give them."
But in Bromley, to the south-east of London, where Ms Harman sent her son Joe, Conservative policy has had an odd effect.
In response to the decision by one school to increase its selection of pupils by ability, all the borough's secondaries have followed suit in a defensive move.
But because all schools will be trying to ensure they get their 15 per cent of able pupils, it will tend to make schools more comprehensive - the opposite effect to that intended by the Government.
"I'm all in favour of every area having a grammar school, but it should be done in a balanced way," says Tory education committee chairman Brian Humphrys. "There should be safeguards to ensure it fits in with the needs of an area. Every school in Bromley has a sixth form, so they need a balanced intake. "
The comment sums up the dilemma at the heart of Conservative education policy. It favours a more market-based approach with selection according to ability, but that makes it difficult to run an effective schools system. And it may, if some of the warnings are correct, turn out to be unpopular with the majority of parents whose children fail to make the grade.
One local Tory activist agreed that his views on education were now in line with those of New Labour.
Could he foresee a time, if his own party continues to move to the right after a general election, when he might be tempted to throw in his lot with the Blairites? "I could be driven to it," he replied.