The correct answer to this open question is the following.
The relationship between Jewish Orthodox and mainstream American culture is one of respect for the American culture and values, but Jewish Orthodox still maintain the strict values that characterized them in their small and private circle.
Jewish Orthodox are grateful to the country that opened its doors and extended and amicable hand. That is why they show the utmost respect for the values and customs of the United States. However, they still living the specific and strict rules that have been with them for millennia. The way men behave and dress, the way they treat women, the role of women in Jewish Orthodox, their intimate circle of relationships, and so many other conducts that characterizes this group of people.