Police in the German city of Leipzig made 211 arrests after far-right supporters vandalised buildings and burned vehicles on the fringes of an "anti-Islamisation" rally.
The unrest came as thousands of activists protested peacefully.
They blame Germany's record influx of refugees for sexual violence against women at New Year's Eve festivities.
Officials say the perpetrators of assaults in Cologne were almost exclusively from a migrant background.
Apparent retaliatory attacks in the cathedral city on Sunday, in which at least 11 people from Pakistan, Syria and Guinea were hurt, were condemned by the government as inexcusable.
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Media captionAnna Holligan reports from Cologne: Anti-Pegida opponents believe the right-wing group is ''exploiting one night of crime in one city for its own ends''
Some 2,000 supporters of Legida, the Leipzig version of the Patriotic Europeans Against the Islamisation of the West movement (Pegida), marched through the city on Monday.
They vented their anger at Chancellor Angela Merkel over her government's open-door policy on refugees.
As the protest took place, a group of far right extremists and football hooligans went on the rampage in the largely left-wing district of Connewitz, police said in a statement (in German).
A window of a Turkish kebab restaurant is broken after a demonstration of by Legida in Leipzip, 11 January 2016Image copyrightAP
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A window of a Turkish kebab restaurant is broken after a demonstration of by Legida in Leipzip
Some 250 rioters smashed windows, burned cars and rubbish bins, and threw fireworks that set a floor of a building on fire, reports say.
Protesters opposing Pegida also held a counter-demonstration in Leipzig on Monday evening.
Left-wing activists vandalised a bus that had been hired by the far-right supporters. The police statement did not say whether any leftists had been detained.
Members of Legida 11 January 2016Image copyrightReuters
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Legida supporters accuse Chancellor Merkel of putting women at risk by allowing the arrival of more than a million migrants in the past year
Protesters opposing the PEGIDA movement (Patriotic Europeans Against the Islamisation of the Occident) attend a counter rally in Leipzig on 11 January 2016.Image copyrightAFP
Image caption
Anti-Pegida protesters, who fear creeping Islamophobia, held a counter-demonstration in Leipzig
The scale of the assaults on women in Cologne and other German cities on 31 December has shocked the country, and police handling of the events has been sharply criticised.
More than 500 criminal complaints were filed over the events in Cologne, with 40% alleging sexual assault.
An official report said the attackers were "almost exclusively" from a migration background, mainly North African and Arab.
Cologne police also made "serious mistakes" in not calling reinforcements and the way they informed the public.
Around 1.1 million asylum seekers arrived in Germany in 2015.
Since October 2014, several German cities have seen large "anti-Islamisation" rallies by Pegida.
The group wants Germany to curb immigration, accusing the authorities of failing to enforce existing laws.