Drys versus wets. Blairites versus Brownites. Commentators love nothing better than dividing politicians into warring and easily defined camps. This has been clear in the past few days since the disastrous Tory result in Eastleigh, with the attempt to neatly divide the cabinet into modernising or traditionalist factions.

As part of this debate, David Cameron's declaration that he wasn't going to be "lurching to the right" was seen as an attempt to burnish his modernising credentials. It was puzzling that this came at the same time as media messaging about cutting the welfare budget further and pulling out of the European court of human rights.

So can Cameron deepen the modernisation of the party while pacifying his backbenchers? Some would argue that was impossible, but that would be taking a one-dimensional attitude to what modernisation was and needed to be.

It's very easy to caricature "modernisers" as being in favour of things like gay marriage, while "traditionalists" are obsessed with things like Europe. But the neat dichotomy of "mods" versus "rockers" doesn't work nearly as well in practice as it does on paper. It's a false choice.

In 2005, it was clear that the Tory party had to change to broaden its appeal and, to some in the popular imagination, that process was all about hugging huskies, hugging hoodies and putting a windmill on your house. In truth, a more important part of the way the Tory party had to change, and still has to change, was that it needed to reach out beyond its traditional base and show that its political role wasn't to defend the interests of the rich and privileged. The most important part of "modernisation" was showing that Conservatism could be compassionate, could help tackle poverty and could be trusted with public services.

This was reflected in the phrase, "Conservative means for progressive ends" – setting out that means that are traditionally regarded as "Conservative", such as market mechanisms and making work pay could achieve ends that are traditionally regarded as progressive, such as fighting poverty, full employment and prisoner rehabilitation.

Despite the neat division beloved of the media, it's actually some of the ministers regarded as "traditionalists", who are following more of a "compassionate Conservative" agenda than some of those labelled "modernisers". There are cabinet ministers who are showing that it's possible to be both Conservative and a social reformer at the same time.

Chris Grayling is widely regarded as a hardline justice secretary, very much on the traditionalist wing of the Tory party. It's true that he has taken a harder line on many issues than Ken Clarke, whose tenure delivered all of the political pain of a "soft" approach, while delivering almost zero policy gain in terms of reduced prison numbers. So the new approach is absolutely necessary for helping to restore the party's law and order reputation. But Grayling's agenda at the Ministry of Justice is some distance from the "lock 'em up and throw away the key" approach of Westminster mythology.

In fact, the new mantra is "more people going to prison, less of them coming back". Grayling's compassionate, Conservative approach means tougher punishment, but combines it with a real focus on helping offenders to go straight. That's why, under his stewardship, the Ministry of Justice is planning to introduce the biggest rehabilitation programme in living memory (progressive ends), backed up by a new payment-by-results mechanism (Conservative means).

You could say the same about Iain Duncan Smith. His past as a Maastricht rebel and his attempts to cut the welfare budget often have him labelled as a hardliner. However, it was Duncan Smith who warned colleagues against using offensive terms like "shirker" regarding welfare claimants and it's clear that he sees his welfare reforms as a means of helping people make the most of their potential.

His visit to the Easterhouse estate in Glasgow instilled in him a determination to fight the human tragedy of poverty. You can see that in the reforms aimed at getting those who are furthest away from the labour market back into work and taking measures to tackle drug and alcohol dependency.

Lord Ashcroft's polling in Eastleigh highlights again that the rise of Ukip is less to do with specific concerns over Europe, immigration or human rights, and actually far more about politicians' failure to connect with voters.

But on tackling social breakdown, Cameron and his cabinet are building a record to be proud of. And on these issues, Cameron connects. This is, after all, the Tory leader who brought the party faithful to their feet to cheer a heartfelt attack on Labour's record on poverty.

A renewed focus on compassionate Conservatism could breathe new life into modernisation. It could also give new purpose to a premiership that sometimes struggles to find a defining mission beyond the painful work of deficit reduction.