"Corbyn can't even snub the Queen properly,” remarked one commentator last week as the embattled Labour leader side-stepped his bowing-and-scraping session with Her Majesty. Mr Corbyn had said he was unable to attend his first Privy Council meeting because of prior engagements.

He is now in a classic double bind: when Corbyn finally does do the obsequious bit he will be accused of hypocrisy. But any equivocation, or even further delay, will likely be seen as unpatriotic and a slight on the monarch.

Compare and contrast with the Right Honourable Nicola Sturgeon who, like her predecessor, Alex Salmond, has had no difficulty bending the knee to the monarchy. Indeed, the First Minister bows and smiles like the best of them.

At the opening of the Borders Railway last month, Ms Sturgeon gushed about the Queen's “dedication and exemplary public service”. Alex Salmond had no problem with the Privy Council and used to lead the singing of God Save The Queen at the Edinburgh Military Tattoo

So how is it that the SNP are not accused of being royal toadies, phoney socialists, fair-weather republicans? The Nats get away with appearing left-wing radicals, complain Labour, while being essentially an establishment party.

It's not just royal flummery and Police Scotland. What have they done about inequality, commentators complain?

This arose again last week following an Oxfam report saying that the wealth of Scotland's top 100 plutocrats rose by nearly 20% last year alone – a faster rate than in the UK as a whole. Meanwhile the numbers of families being referred to food banks in Scotland increased to 70,000.

Why, after eight years, has the SNP not addressed these deep social divisions? The educational attainment gap still shames Scotland's claim to be a country of the “common weal”. A smaller proportion of working-class children go on to university than was the case 10 years ago.

At their conference in Aberdeen this week – the biggest in SNP history – Nicola Sturgeon will be at pains to reject charges that her government has lost its radical edge. She'll say she has “staked her reputation” on closing the schooling gap between richer and poorer areas in Scotland.

She'll attack the hypocrisy of an Eton-educated prime minister preaching equality of opportunity when he leads a cabinet stuffed with public schoolboys.

On poverty and redistribution she'll say the Scottish Government's hands are tied by Westminster. Hasn't even Gordon Brown now admitted that the vow has not delivered the promised powers to increase welfare benefits in the Scotland Bill?

The Scottish Government lacks the power to slap a new 50p top rate of income tax on billionaires (at least not without increasing all the other bands too under the new Scottish rate, which comes in next year). Scotland won't have full tax-varying powers until the Smith reforms are implemented – assuming they are – in about 2018/19.

Meanwhile, the Scottish Government has taken steps to mitigate the UK Government's more objectionable benefit reforms, like the bedroom tax. Sturgeon will lambast Labour MPs for abstaining on the Tory welfare bill in Westminster in July and Labour front-benchers for flirting with Tory victimisation of immigrants.

As for housing, the right to buy has been abolished in Scotland and the Scottish Government is committed to restarting council house building. Scotland has one of the few gender-balanced cabinet governments in the world.

The SNP remains the only major party going into the 2016 Holyrood elections committed to abolishing nuclear weapons – though it has no powers to do so. The First Minister will challenge Jeremy Corbyn to join the SNP MPs in opposing the renewal of Trident in next year's Westminster vote.

Prescription charges have been abolished, university tuition is still free, personal care has been funded. Of course, many in the Labour Party dismiss these as “middle-class benefits”.

But it's fair comment that most of the SNP's achievements are modest reforms at best that have done nothing to challenge the established structures of wealth and power. There is a mismatch of rhetoric and reality.

The SNP has presented itself as a left-wing alternative to the Labour Party, but in office it has not so far behaved very differently to the previous Labour-Liberal Democrat coalitions.

It's not just on social justice. Labour claim that the SNP's radicalism on the environment is skin-deep; that it is in the pocket of the fossil fuel industry and ready to sell out on fracking to the Ineos boss, Jim Ratcliffe.

Last week, the Scottish Government was under fire from trades unions, the Greens and the left-wing RISE party for privatising water services to schools and hospitals. The infrastructure secretary, Keith Brown, insisted that he had to award the contract to Anglian Water because of competition rules laid down by the previous Labour government.

As with the possible privatisation of Caledonian MacBrayne ferry services, and the awarding of rail franchises, critics say this SNP Government has been far too willing to allow its hands to be bound. Is it really committed to public ownership?

There is a fundamental issue here. Unlike RISE or even the Greens, the SNP is not an ideological socialist party that regards nationalisation as a good thing in itself. Many prominent SNP MPs and supporters are businessmen and women. I spoke to one last week who sells aeroplanes to African countries.

Then there is the former Westminster business spokeswoman, Michelle Thomson, whose property speculation went unnoticed until exposed by a Sunday Times investigation. She has resigned the party whip and sits as an independent. A more left-wing party might have noted earlier that Ms Thomson was deeply involved in the property business and asked some searching questions.

Mind you, there has been no shortage of property speculators on the Labour benches in Westminster. Even the left-wing Labour MP Michael Meacher, owned no fewer than seven houses at one point. The Thomson affair is embarrassing to the SNP, but it has very little direct political relevance to its moral war with Labour.

Social democrats like Nicola Sturgeon accept that a buoyant private sector is the only way to generate the revenues necessary to finance decent public services.

The SNP has done well in recent years by adopting many leftish Labour themes – like free higher education – abandoned by New Labour under Tony Blair. But the SNP's electoral support doesn't depend on any consistent commitment to socialist policies.

The SNP has traditionally – like all nationalist parties – been a coalition of left and right. Look at some of the party's key financial backers, such as Brian Souter of Stagecoach or George Matthewson, former head of the Royal Bank of Scotland.

Indeed, the SNP as a party of the left is a recent development and it may already have served its purpose. The party's over-arching goal is, after all, independence and that requires mobilising rich and poor to the cause of self-government. To win the next referendum, the SNP must secure the votes of middle-class home owners and pensioners who voted No last year.

The demographics of nationalist support continue to amaze. According to last week's TNS poll, the SNP now has 68% support among voters between 16 and 34, which means it is the party of the future. For most of these voters, the SNP is quite left-wing enough. Indeed, whatever Nicola Sturgeon says is good enough for them.