Using the American Time Use Survey for the years 2003-18 we compare the allocation of time of native men and women married to immigrants with that of their counterparts in all-native couples. We find that when intermarried to a native some immigrant women pay an assimilation price to the extent that, compared to native women in all-native marriages, they work longer hours at paid work, household chores or both, while their husbands do no extra work. In some cases they work an extra hour per day. Immigrant men don't pay such price. Some work 34 minutes less at household chores than native men in all-native marriages, while the native women who marry immigrant men seem to pay a price relatively to what their situation would be in an all-native marriage. An explanation based on the operation of competitive marriage markets works for immigrant women but not for immigrant men. Traditional gender-based privileges may allow immigrant men to prevent native women from capturing a price for the value that intermarriage generates for their husbands. Such 'male dominance' scenario also helps explain why immigrant men married to native daughters of immigrants from the same region get more benefits from intermarriage than other immigrants.