I was asked this a while ago, and I wanted to publish the answer here as well. There are plenty of labels I identify with religiously—polytheist, Pagan, Hellenist—and I have found that the labels I identify with are linked closer to the experiences of others than my own. I label myself in a way that is understandable for others; I know who and what I am, I have no need to label myself just for me. When I am in the company of people who will have most likely never heard anything about Hellenism or even Paganism, I stick to ‘Hellenist’. At least that way, I can explain right away what I believe in. With other Pagans, I identify as ‘Pagan, Hard Polytheist—subset: Hellenist’. With other Recons, I identify myself solely as ‘Hellenist’. Heck, with my parents, I’m just ‘religious’, that’s hard enough for them to agree with.

Labels, in my opinion, have their value placed upon them by those outside of the label; as much as a group might feel they have claimed and formed the meaning of a label, in truth it will always be defined by those outside of it. I have gone on record saying that I think Hellenismos is part of the Pagan banner, not so much because we define as such, but because the outside world sees us as part of it—so as for your question; I think nothing does; I think Hellenists are labelled Pagans.
There are many ways to look at our inclusion in Neo-Paganism, however: one could make a case for being outside of it but also within it. Neo-Paganism is so undefined that it can encompass any religion or tradition looking to be a part of it; as long as it is at its core non-Abrahamic, it can qualify—even though that line is muddled and not very useful to begin with.

I think Hellenismos is moving away from Neo-Paganism slowly but surely; we’re growing as a religion, and we are unifying in a way that was previously unheard of. I feel this has a lot to do with the rupture that went through the Neo-Pagan movement as a whole a while back with the whole ‘Pagan vs. Polytheists’-thing. While that debate seems to have mostly died down now, it did leave a desire in many Reconstructionists to focus more on practices and traditions similar to ours instead of the more nature-based other half of Neo-Paganism. I see more calls for Recon events now, or for Recons to have more (or a separate) say in events already hosted. Recon blogs are more vocal now, and we are becoming far more mainstream.

That said, the line between the ‘classical’ Neo-Pagan traditions like Wicca, Neo-Wicca, shamanism and even Druidism and the Recon faiths is thin; it’s far more likely these days that modern Recons came from one of these traditions than straight into Reconstructionism itself. Many of us—myself included—wandered through many forms of Neo-Paganism before ending up here and for some the ability and/or desire to shake those previous practices was greater than others. The same holds true if you came here from Christianity or any of the other religions known to man.

At the end of the day, ‘(Neo-)Pagan’ and ‘Hellenist’ are simply labels. It’s a way for me to quickly convey to others who I believe in and what I do. That said, the usefulness of both labels depend on the knowledge these ‘others’ possess, and I have zero influence over that.