A Cabinet reshuffle can often seem like a grand game of musical chairs – a political ballet with plenty of noise but little significance. But while the resulting changes often seem more cosmetic than consequential, for David Cameron, next month’s expected switch-around is likely to be a genuine test of his leadership.

The main challenge he faces is the need to appease agitating backbenchers on the Right of the party, while simultaneously keeping the Tories grounded in the political centre. The revolt over Lords reform has weakened his position and emboldened the Conservative Right-wing. A show of force – of cajoling and whipping backbenches into line – has so far proved inadequate. Cameron has only one option left: concession.

As such, one expected reshuffle is likely to concern minister for Europe David Liddington, who has felt the ire of the party’s Right by being seen as “insufficiently Eurosceptic”. Another possible casualty is party co-chair Baroness Warsi. Unpopular with the party’s Right and struggling in an ill-defined role, her recent breach of the House of Lords code of conduct – a misdemeanour that stuck, even though she was ultimately cleared of expenses fraud over accommodation claims – may provide Tory HQ with an easy opportunity to quietly move her on.

But herein lies the problem. For a government that has been heavily criticised for being “unrepresentative”, a demotion for Warsi will surely provoke awkward headlines in certain areas of the press. The problem has been exacerbated by Louise Mensch’s decision to resign. Mensch had proved popular with the electorate, and a strong performer in select committee hearings. She was widely tipped for promotion, and leaves Cameron with a familiar problem – a lack of high-profile, female colleagues in the party’s upper echelons.

Similarly, removing Liddington and appeasing the right of his party may be a necessary evil for Cameron. But given the limited success of his “detoxification” of the Tory brand, there are concerns that, in doing so, he will alienate floating voters put off by further dog-whistle politics.

This says nothing of the simmering resentment that has been bubbling between the Coalition partners, and now threatens to boil over. A return to the front bench for David Laws may go some way to reduce tensions – but the cracks now run deep.

These are dangerous times for Cameron, and it remains to be seen whether a reshuffle can effectively position him as both a protector of traditional Conservative values and a champion of more socially liberal policies such as gay marriage.

At the very least, a reshuffle needs to demonstrate a reassertion of his authority – not a reduction.