The Conservatives have accused deputy Labour leader Harriet Harman of re-opening the class war.
It followed her speech to the TUC in which she said the gap in opportunities between rich and poor must be tackled.
She told the BBC she wanted everyone to "get a fair crack of the whip" whatever their "socio-economic class".
Shadow Commons leader Theresa May said that trying to focus equality efforts on class and background was "outdated and distracts from the real issues".
In her speech to the TUC annual conference Ms Harman said that "equality matters more than ever" and "is necessary for individuals, a peaceful society and a strong economy".
Ms Harman's comments came after Prime Minister Gordon Brown said in an interview with Monitor magazine that "social mobility has not improved in Britain as we would have wanted".
They will be seen by many as an attempt by the government to rally the unions to Labour, after widespread criticism over the level of public sector pay and demands for a windfall tax on energy firms' profits.
Ms Harman, who is also Commons leader and minister for women and equality, said that "great progress" had been made on tackling inequality on the basis of things like race, gender, sexual orientation and disability.
But she said more needed to be done to tackle the inequality of opportunity between "the rich and poor" and "the north and the south".
Although she dropped the word "class" from the text of her speech when it was delivered, she later denied on the BBC's World at One that there was any taboo about using the word.
She said: "I think that socio-economic class and background is a kind of policy description and I think most people think about it in terms of family background - what sort of family you come from and where they come from.
"Those are the things that people know still underpin some inequality that we want to tackle."
She said examples were the fact that a rich man can now expect to live longer than a poor woman, and that a less able child from a richer background will overtake a brighter child from a poorer background by the age of six.
Ms Harman announced that the government's National Equality Unit would be headed by Professor John Hills of the London School of Economics.
Not quite class war then but a hope to redraw the dividing lines with the Tories
She said: "The robust evidence base that the panel will produce will help us properly target measures to address persisting equality gaps and build on the good work that we have already done."
Ms Harman accused the Conservatives of being "false friends of equality" and of "sidling up to the unions".
For the Conservatives, shadow leader of the Commons Theresa May said: "I am astounded that Harriet Harman is dismissing the equality issues around race and gender."
She added: "I also find it surprising that she should raise issues of social equality when she's part of government that has been in power for over 11 years, presiding over a 900,000 growth in the number of people living in severe poverty and over a country that has the lowest social mobility in the developed world.
"Labour has made poverty more entrenched and returning to the class warfare rhetoric of 20 years ago is neither helpful nor realistic."
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