The Norman invasion is often taught as a distinct stage in English history, one that caps off the Saxon period, but the Saxon culture continued to exist after the Normans, indeed a guerrilla war prompted the Normans to treat murder against their number with the harshest severity.
Even after the period of instability there still seems to be a disconnect between the Saxons and their Norman overlords, I've had people tell me before that England was merely a resource pool for the defence of the French territories, and a famous saying goes "English on the field, French on the table." referring to the disconnected words for livestock and their meats, and right the way up to Civil War there was a notion that the Normans were still the corrupted upper class gentry, and that they should be overthrown by rightful Saxons.
My question is; is the English culture we know today Norman with Saxon influences tied into it, or Saxon with Norman aspects?
It has been pretty firmly established that the conquering Norman element managed to hold its own and gain linguistic converts, for roughly the duration of the Angevin Empire. Richard the Lionhearted hardly spent any time in England at all, and his power base was concentrated almost entirely in Normandy and the Aquitaine.
This changed with his immediate successor, the infamous King John, who was in many respects, the first 'English' King. In the sense that owing to the loss of his continental territories, he was forced to focus on England as his power base. During his reign, tax revenues from England skyrocketed. A simultaneous result of King John's rule, was the disastrous Anglo-French War of 1213-1214 which cemented the loss of Normandy (already lost ten years earlier).
From this point on, a Norman could not hold both estates in Normandy, in service to the Capetian King, while simultaneously holding English estates. With only one meaningful exception (the famously popular tourney knight/landowner William Marshal), all other nobility had to 'pick sides' as it were.
Those who predominantly held lands in France, forfeited their English estates. Norman nobility who held most of their estates in England, forfeited their French estates. The ordeal of the early 13th Century Anglo-French Wars would leave a wound in the English psyche among the nobility, to reclaim their lost inheritances. As all know, this eventually culminated in the Hundred Years War (by which point, England was thoroughly English, even if French remained the favored language of the nobility, as it would be in most of Europe for centuries).
To put it another way:
The Angevin Empire of the 12th Century saw many continental elites given estates and authority in England as a reward, and there was a clear move by English subjects to adopt continental culture in order to secure their advance.
The truncation of the Angevin Empire into 'England', which remained almost exclusively the case for the next hundred and some odd years, greatly arrested the drift of England into the continental orbit. Instead, it would form its own nascent state on its own terms, and would direct most of its energies (initially) to conquering the rest of Great Britain.
By the time the English returned to the continent to reclaim their inheritances, they had become a very different people - and none of their conquests, outside Calais and Gascony, were ever truly secure. Calais and Gascony were secure owing to the overwhelming financial incentives of the Flemish and Gascon burghers to maintain their English trade routes, and less from a shared culture.
After the Hundred Years War, England's focus once more turned to Great Britain, and it would be the next two centuries of unrest, civil war and good old fashioned un-civil war (such as the Cromwell campaign in Ireland) that would put the definitive stamp on Englishness. Right up until the 20th century, a great majority of Scots and Welsh considered themselves 'English' as well. The rebirth of Scottish and Welsh national movements is a phenomena of the last fifty years. Before that, it was considered as unrealistic as those wanting California to secede.
So there you go. The Normans conquered England and remained extremely dominant and extremely focused on continental affairs for the next ~150 years, which saw a proliferation of both the French language and customs (such as tournaments). The vast majority of all French loan words were acquired in this time.
The loss of all French territories after ~1200-1214, severely reversed the pendulum, and by the time of the Hundred Years (which 'officially' kicked off around 1337) the English had their own fully separate identity, and by then, had no real meaningful ties to 12th Century Norman culture. (No more than modern Americans have much of a tie to the Puritans).
Of course, for centuries thereafter, the inequality of wealth and land ownership, concentrated nobility and privilege among families with mostly fancy Norman surnames. But they were all quite English.
Also. Keep in mind that I've referred to 'French' culture here quite broadly. There were dozens of distinct French cultures (such as Norman, Burgundian, Gascon, Poitevin, Breton etc) that were all very distinct. It took about six hundred years, starting from the Albigensian Crusades and ending with Napoleon, to enforce 'Frenchness'.