Cassidy has been described as a turbulent priest by John Major, and has suspiciously (for the City) close ties with the Labour Party leadership. Mayhew came within a whisker of becoming Tory candidate for Hampshire North West, one of the safest seats in the country. Now add that Mayhew is the first woman to become chairman of the Corporation's Policy and Resources Committee - the equivalent of leader of the council - and the first foreigner to do so (she's a New Zealander), and the contrast seems complete.
But it ain't quite so. For a start, both are lawyers and tough political (with a small "p") operators, and know that to fend off the philistines who want to abolish the 1,000-year-old corporation, they have to reform it. In fact, the philistines (aka elements in the Labour Party) are not the real problem: Cassidy's power came as much from his personality as his title and there are those in the Corporation who would like some of that power. "Judith Mayhew is taking over a can of worms," one insider says. But despite her quiet-spoken neatness, she is determined and ambitious: she must have a good chance of sorting the can out.
The Corporation of London is a strange beast: a mix between the Millennium Commission and the Confederation of British Industry, wrapped in 1,000 years of anachronisms. Mayhew is not a councillor, she is the first line of a Gilbert and Sullivan song: a Common Council Man of the Court of Common Council. She sits in the lower chamber; the upper chamber is the Court of Aldermen, from which the Lord Mayor is drawn.
Mayhew is fascinated by such things. "My sense of history is highlighted because I learned about it from a distance," she says. "New Zealand is a young country so we find the continuum here quite amazing."
The modern City is another source of quirkiness. It has 5,000 permanent residents and one primary school, yet it has a daytime population of 250,000 and is one of the richest spots on Earth. The Corporation owns a third of the buildings, which makes it enormously wealthy. It is a major philanthropic force in London, which is why the Labour Party dropped those abolition plans. Yet the Corporation has come close to being rate-capped because the Government allows it to spend only pounds 70m of the pounds 700m it raises in rates.
Mayhew says the Corporation's main role is to ensure the City maintains its dominance in the financial markets. It works closely with the Bank of England on technical matters and takes the lead where the Bank cannot be seen to be active. She sees one of her principal tasks to be ensuring that London becomes the centre for trading the new European currency.
The combination of infighting and outfighting will keep Mayhew busy. Unlike other council leaders, the Chairman of the Policy and Resources Committee does not get paid. Fortunately Wilde Sapte, the law firm she works for, has taken an enlightened view of the benefits such a high- profile employee can bring. "It's prepared to give me as much time as I need," she says.
Judith Mayhew was brought up in Dunedin on the south island of New Zealand. She did a law degree at the University of Otago, where she overlapped with (but did not know) Paul Beresford, who became leader of Wandsworth council and is now a junior minister.
She became a legal academic and at the age of 24, in 1972, took a job at Southampton University. "We all came to England," she says. "The difference was I didn't go back." That was partly because she married a barrister on the West of England circuit. They bought a house in Hampshire; she still has the house - but not the husband - and grows old English roses there as a hobby.
In 1976 she moved to London to take up a job at King's College and bought one of the first private flats in the City, on the Thames near Southwark Bridge. She still lives there, which means she can walk to work and to the Guildhall. She didn't became a practising lawyer until 1989, aged 41, when Wilde Sapte asked her to join as an employment lawyer and its director of training. "I had six months of total terror, but I adjusted," she admits.
Ten years ago the residents in her block asked if she would stand as a Common Councillor. She did and became ever more involved in the Corporation's works. As chairman of the Education Committee, she pioneered adult education vouchers and says her greatest achievement has been to refocus training and education in the City after the break-up of the Inner London Education Authority.
She has been Cassidy's deputy for four years. They say they work well together, though others within the Corporation suggest the relationship may not have been so smooth. Her decision to go for a Parliamentary seat last year ruffled feathers because it exposed her as a political animal in a theoretically non-political body. "I stood because it's always been a dream to be a national politician," she says. "But I don't let party politics intrude in the Court of Common Council."
Her instincts have always been Tory but she didn't become an activist until the run-up to the 1992 election when Labour was promising to abolish the Corporation. She is not, she says, a hardliner, pointing out that one of her New Zealander chums is Brian Gould, the Labour man who has now returned home.
Mayhew defends the Corporation against charges that it takes a high-and- mighty attitude - the security cordon that restricts traffic through the City is particularly unpopular elsewhere. The cordon is necessary, she says, to keep foreign banks there and to oil the City's wealth-creating mechanisms: "Many of us spend a lot of time in taxis going to meetings because that's the way the City works."
But she does believe there must be an increase in democracy. Only residents and firms with partners can vote for councillors; she would like all corporations to have votes, based on rateable value and maybe numbers of employees.
She is involved with UK-New Zealand bodies and believes that the two countries are getting closer. "We've been through a difficult period when Britain looked to Europe and New Zealand looked to Asia. But we've now realised that local commercial links can be used for the other country's advantage."
One of Cassidy's strengths is his close links with the media. Despite her mobile phonelessness, Mayhew is likely to repeat the trick. She presented a television programme on the first woman lawyer in the Empire, a New Zealander called Ethel Benjamin, and says she loves doing television. With her precise lawyerly mind, she should be pretty good on the Today programme, too.
Just as Cassidy is pitching to be Mayor of London, we can be fairly sure that Judith Mayhew's public career will not finish with the Corporation. She has already shown her interest in Parliament and might reasonably expect to become an Alderman and perhaps Lord Mayor. Meanwhile, she has that can of worms to deal with.