How did the twins Romulus and Remus help shape Roman culture?

Romulus and Remus are two figures of Roman myth and the legendary founders of the city of Rome. As the legend has it, Romulus and Remus are the twin sons of Rhea Silvia and Mars, the god of war. When the twins were born, Rhea's uncle (who stole the throne from her father) stole the babies away and ordered them to be drowned in the Tiber River. Luckily, the infants floated to safety and were looked after by a wolf and woodpecker, then taken in by a shepherd. As young men, the twins get into a fight with some of the king's men, and Remus is captured. Romulus kills the king, who he does not know is his uncle, and rescues Remus, restoring their estranged grandfather to the throne. 
The twins then decide they wish to found their own cities, though they disagree about the location. Romulus decides to build his city on Palatine Hill, and constructs a wall to secure the location. Remus thought building a wall was a silly idea and even tried to prove how ridiculous the wall was by jumping over it. In anger, Romulus killed his brother, and named the city for himself—Rome. This is the foundational myth of Roman culture and how the city got its name.
The myth of Romulus and Remus explains the name for the city, and further details help provide a mythological or historical ground for some of the geopolitical and socioreligious values of Roman culture. It is said that after founding his city, Romulus offered asylum to those who had been kicked out of other places, including criminals, the sick, and fugitives. The population of Rome was overwhelmingly male, and to obtain some brides for his citizens, Romulus invited women from the native, neighboring tribes of Sabines and Latins to come to the city for a festival. There, the women were captured, assaulted, and forced to marry the Roman men. This part of the myth offers a "justification" or societally-grounded inspiration for the continual capture, assault, and forced assimilation of many of the tribes surrounding Rome. Despite forced assimilation, these cultures were heavily incorporated into the new Roman society—for example, many Roman gods are parallels of those worshiped by the Sabines.
In response to the capture and abuse of women and children, these neighboring societies declared war on Rome, but were ultimately defeated. When defeated, their territories were forfeited to the city of Rome, and here we begin to see the development of the Roman Republic. Rome began to include surrounding lands outside of the city wall and transplant "Roman" culture to the farthest reaches of the Republic. 
As we can see, the foundational myth of Romulus and Remus provided not only the name for a city and culture, but a cultural identity grounded in expansion, conquest, and social dominance. Their actions described in the foundational myths of Rome fostered an attitude which grew into the Roman Empire, which included much of Europe and North Africa.
https://www.ancient.eu/Romulus_and_Remus/

https://www.britannica.com/biography/Romulus-and-Remus

https://www.britannica.com/place/Roman-Republic