AFTER springtime rulings friendly to gay rights and Obamacare, the Supreme Court is likely to swing back to the right when the justices dust off their robes and return to work on October 5th. Of the roughly three dozen cases they have already agreed to hear in their new term, as well as a few that seem destined for the docket, a handful could shake America’s political landscape.

A blow to public-sector unions looms in Friedrichs v California Teachers Association. The Centre for Individual Rights, a libertarian law firm, is representing ten teachers in California who object to the union fees they must pay. No one can be forced to join a union, but the Supreme Court ruled in 1977 that teachers, police officers and other public employees can be compelled to pay a “fair share” fee equivalent to membership dues. The justices reasoned then that since unions negotiate for benefits on behalf of a whole sector, such a charge prevents those who refuse to join a union from free-riding on those who do.

But the teachers say these fees violate their freedom of speech. In their brief to the justices, Thomas Jefferson plays a starring role: “To compel a man to furnish contributions of money for the propagation of opinions which he disbelieves,” the Founding Father is quoted as saying, “is sinful and tyrannical.” If five justices agree that Jefferson’s sentiment applies in this context—and several hinted in a ruling of 2014 that they do—then public-sector unions could face a crisis of funding and a steep decline.

Fisher v University of Texas offers the justices a chance to overturn precedents upholding race-based affirmative action at public universities. In 2008 Abigail Fisher applied to the University of Texas at Austin, where admissions officers take note of an applicant’s race as part of their effort to improve student diversity. Ms Fisher, who is white, was rejected. She contends that the university’s policy constitutes racial discrimination and is therefore banned by the 14th Amendment. In 2013, after the Fifth Circuit court had rejected Ms Fisher’s claim, the Supreme Court told the lower court to reconsider its ruling. A year ago the Fifth Circuit again upheld the preferences, spurring Ms Fisher’s second petition to the justices. This time, many observers expect the court to rule in her favour and possibly to abandon its decision in Grutter v Bollinger, in 2003, which allowed limited consideration of race in higher education.

In Evenwel v Abbott, the court will clarify its ruling from more than half a century ago that states must apply the principle of “one person, one vote” when drawing up electoral districts. Texas, like most states, uses the total number of people as the denominator when drawing district lines. But Sue Evenwel and Edward Pfenninger, Texans who live in districts with many eligible voters, complain that this dilutes their voting power. They have less representation, they say, than voters in districts with large numbers of felons, children and aliens—people who are ineligible for the franchise. If the court agrees with the plaintiffs, all 50 states will need to rip up their district maps and draft new ones. Electoral power will shift from cities to rural areas, where higher concentrations of eligible voters, and Republicans, live.

Few hot-button issues will escape review: racial bias in jury selection, criminal justice for juveniles and the death penalty are all on the calendar. The justices have not taken an abortion case since 2007, but this year there is a good chance that they will hear one or more challenges to stringent new rules that aim to shut down abortion clinics in Mississippi and Texas. At issue is whether these regulations place an unconstitutional “undue burden” on a woman’s right to have an abortion. Another related case may also get a hearing, this time involving religious non-profit groups who are unhappy with the Affordable Care Act’s birth-control mandate—and with the Obama administration’s attempt to exempt them from the requirement.