Perhaps the most important facet of relations between Romans and "barbarians" is that this was not a binary distinction and there was room for significant overlap between the two. Indeed many Germanic groups were invited into the Empire to serve as foederati, and others that remained outside of the Empire officially were still able to maintain quite close relations through trade but also military service in the Roman state. It is worth pointing out that many of the Germanic groups that occupied former Roman territory attempted to preserve many aspects of Roman life. Theodoric in Italy is particularly renowned for his efforts at preserving Roman life in Italy, even in the face of his own conquests. Indeed in places such as Italy and Southern Gaul, Roman law survived, in addition to substantial Roman estates, infrastructure, and civic life. There was clearly some level of tolerance, and in some cases perhaps respect and deference to Roman traditions among the newcomers.

However it goes deeper than this. Many Germanic men would serve in the Roman army and then return to their homes, we know this because of the burial of Roman military belts and other symbols of office found in parts of the Germanic homelands that were not a part of the empire. This indicates, at least to figures such as Guy Halsall and Peter Heather, that "Germanic" and "Roman" were not fundamentally opposed identities. Indeed many prominent Romans of late Antiquity were of Germanic descent, most famously Stilicho, the de facto ruler of the the western portion of the empire for some number of years.

The majority of the migratory groups moving into Roman territory had, at least officially, adopted Christianity, even if a heretical variant, Arianism, was quite popular for a time among these groups. In England the Angles, Saxons, and Jutes were indeed pagans but they seem to be the exception in many ways. Their adherence to Arian Christianity was a sore point of contention between these groups and their new Roman subjects, as well as in their relations with powers such as the Byzantines.

It would be far too dismissive to suggest that the newcomers in England and Italy "exercised an harsh right of conquest" as compared to the more mild treatment of the Franks. Indeed such holdovers of the 19th century romantic notions of the "Voelkewanderung" are pretty thoroughly discredited in academia today. For example, in England, the Angles, Saxons, and Jutes were also liable to intermarry and assimilate the native British into their own communities as opposed to just straight up slaughtering them all and taking over their territory. Indeed, Robin Fleming in Britain After Rome suggests that the initial migrations into England by the Angles, Saxons, and Jutes and other Germanic groups were actually quite peaceful with little evidence of the endemic warfare proposed by Gildas, our only roughly contemporary written source. While certainly not a universally adopted idea, it is interesting that the archaeology of the "conquest" portrays it as anything but.

Fleming argues that instead of the old conquering bands of warriors carving out their homes at the expense of the native British, instead we should imagine essentially farmers uprooting to a new land in the face of economic collapse. The only reason they appear to be less Romanized than their continental bretheren is that the Roman economy of Britain had entered a state of free fall and had been abandoned by the Roman state. In the face of this clinging to Roman identity lost out to the newly created, and partially syncretic, identity of the Anglo-Saxons.

If you're more specifically interested in specific customs, that is a bit of an open question. Were inhumations found in "Anglo-Saxon" contexts copying Roman styles of burial or bringing their own style from their homelands, there is after all no one mode of burial that is unchanging in any society, and attempts to categorize settlements as British or Germanic based on their deposition of grave goods or burial method are inherently fraught and do not hold up to scrutiny. Instead of a false dichotomy between Roman and Barbarian we should instead be looking at this as a complex spectrum.