The culture of Rohan is an unusual one. The entire setting of LotR and Middle-earth is essentially based on Norse myth--you know, that whole "Ring" business. In the same way that 'Beowulf' is a story about Norse-men told from an Anglo-Saxon perspective with interweavings of both cultures, Rohan is an equally interwoven glimpse into both Anglo-Saxon and Viking worlds.
In 'Beowulf', Norse names are anglicized by the poet, and certain aspects of Norse culture are emphasized based on Anglo-Saxon philosophies and ideals. The story is about Vikings, but the story is actually an English one.
Rohan's situation is similar, since many of the names and ideas came from 'Beowulf' itself. The same pattern remains: Norse names and ideals, anglicized from an Anglo-Saxon perspective.
Certain details in the story, for instance Théoden's use of "sister-son" to describe his nephew, or the hilltop 'mead-hall' of Meduseld in the capital of Edoras, are certainly Norse in origin. The Rohirrim were also originally "North-men," having only recently emigrated to Rohan.
On the other hand, their language has been anglicized. "Éowyn" comes from the runic alphabet, an Anglo-Saxon creation, as do many other names. It's almost as if ancient Finland were plucked off the map and synthesized into Old English culture, as Tolkien's history concerning Rohan revolves around the Old English language, not Old Norse.
The Vikings were known for their ability to adapt--not necessarily to assimilate themselves to a culture, but they were very good at using bits and pieces of other cultures to benefit their own way of doing things. Rohan is similar, no doubt having adopted many elements of both Dúnedain and Gondorian cultures (at different points in time) to create their own distinctive, partially-nomadic culture, while maintaining their original "Norse" ideals.
Misunderstood as barbarians and ruthless butchers, the Vikings were actually an efficient group of people who were simply looking to establish themselves in the world. Historically, they were one of the most consistently embracing and accepting groups. To fight a war over religious or cultural differences was regarded as impractical (how very civilized indeed).
We don't notice this characteristic in Rohan, as the country is not particularly enterprising (it doesn't have time to be, anyway, since so much of what we read regards Rohan's defending itself against invasion). And this is where Anglo-Saxon comes screaming in at full-speed.
England sits rather in the middle of things, and has certainly had its share of invasion and resettlement over the centuries. The land of Rohan is no different. Sitting smack in the middle of Tolkien's LotR setting, it suffers onslaughts from all borders (some of them bad, some of them good). While adapting to aspects of other cultures that are deemed beneficial, Rohan remains quite resistant to change--understandable considering the plagues of attacks its received over the years.
If we look at the specific details of Rohan's culture, we see more Anglo-Saxon influence. Descriptions of shields, spears, swords and helmets are clearly more Anglo-Saxon. Also, the Vikings were famed sea travelers, while Rohan is a landlocked nation (though, in this case, the Viking culture of the "longship" could easily be paralleled to Rohan's culture of the "horse").
It seems, with Rohan, that Tolkien has placed an Anglo-Saxon culture in the middle of a Middle-earth based on Norse myth. Since Tolkien himself has described the 'anglicization' of many name and story elements within LotR, though, we can conclude that, like the poet of 'Beowulf,' Tolkien is merely telling the story of this world with his own histories and intentions in mind. Perhaps, then, Rohan is his own England, beloved and at the heart of his world, yet always struggling to define and defend itself.