The X-box probably would not be very much good in the long term anyway. As I understand it, it could play, but but not record, HDV discs. But as you would be aware, no more HDV burners are being produced since Blu-Ray won the war a few months ago.
Unfortunately, though, one way or another, you will need to spend money to be able to watch your HDV on your HDTV. The only way which does NOT involve expenditure is the simple one of editing your HDV project as normal in that format, and when it is done, exporting it back to your HDV camera in that format to a new mini DV cassette (Share > HDV Recording); connecting your camera by its HDMI connection (I assume it has one) direct to your HDTV and in effect use the camera as a high definition player. It works very well, and in fact as I write, I am looking at the latest one I did this way.
Unfortunately, that will not work if your camera is, like mine, a Canon HDV model such as the HV10, 20 or 30. Other brands are fine. The really curious thing is that this conflict was recognised after VS10 came out and Corel/Ulead issued a specific Canon patch to fix it in VS10+. But somehow or other that code does not appear to have been incorporated in VS11/11.5+.

I have tried exporting to my Canon HV20 using Adobe Premiere Pro 3, but that also does not work.
In fact, the only work around I have found presumes you have VS10+ already. I reinstalled that on my computer, as well as VS11.5+. I installed the Canon patch for VS10. I edit my HDV in VS11.5+ (though I could also do so in VS10+), then save the project in VS10 format. I open that project in VS10+. I connect my HV20 via firewire and turn it on in PLAY mode, and then go to Share > HDV Recording. And it works! Wonderfully. I would like to think this defect is corrected in the forthcoming VS12, but I am not holding my breath...
The other options -- the ones that cost! -- involve lashing out (considerably!) and buying both a Blu-Ray burner and player, plus the very expensive blank Blu-Ray discs, and actually burning your project as a Blu-Ray disc.
But there is a half-way house: a hybrid AVCHD disc which involves converting your edited HDV to high definition AVCHD, but burning that in its high def format to a standard DVD. But the catch is that you require a player than can play such discs. Again, that involves Blu-Ray and Blu-Ray players are still rather expensive, and in addition, not all of them are rated to play such hybrid discs.
But the Sony PlayStation 3 is one such player (not the X-Box, I am afraid). And the PlayStation 3 (PS3) adds a number of options. First, it will play hybrid discs. But the catch with them is that if (like me) you want the best quality playback from your high def original video, then you have to, understandably, use the highest quality settings for AVCHD. And that means that you can only squeeze about 20 minutes of high def video onto a single layer standard DVD. That's OK for me since I pay only about 25 cents per DVD blank (when the cheapest Blu-Ray blanks here are literally 100 times that price -- each!) VS11.5+ (and the recently released Movie Factory 7) also allow you to build a proper menu for such discs, unlike a number of other editing packages at the moment.
But the PS3 is even more versatile. I have mine networked to my computer, and the PS3 also connected via HDMI to my 46" HDTV. Using Windows Media Player or Nero Media Home as the server, I can thus play all my edited video (in any format, not just HDV; plus still images and music) direct to the HDTV via the PS3. Top quality and wonderful!!
But wait! There's more!!

(Sony should be paying me!

) You can also transfer your edited HDV to either a suitably large USB stick or even an external hard drive, and plug either of those into the PS3 and play your edited HDV back on the HDTV that way. The only catch is that the PS3 can only see drives formatted with the 'old' FAT32 system, not the current NTFS one. This in turn means the maximum size of the individual files it can play is 4 GB (a FAT32 limitation). For HDV format, this is again around 20 minutes of video, which luckily, 95% of my projects are -- I don't like to bore people (too much...

)
And finally, you can burn an edited HDV clip slightly larger than 4 GB to a standard DVD (i.e. max. 4.3 GB) but as an archive/storage disc, not a video disc. And when you put that in the PS3, as long as it is in a folder labelled VIDEO (in upper case!), the PS3 can see it and play it in all its high def glory as well...
All of my personal options above, of course, mean that you have to invite friends and family to your home to see your masterpieces in high def. But until such time as the price of Blu-Ray burners, players and discs come down considerably, and all your family and friends have at least a Blu-Ray player, your options are in any case limited to either standard DVDs (still) or one of my options...