I want to display a 3D model on a website where people can rotate it in any direction. Years ago there were several different plugins for this, like Cult3D. What are people using these days? I don't want to use rendered sequences to simulate rotating a 3D image, but want to display the actual 3D model. Lastly, the model needs to be textured.

Any suggestions?

Thanks.

WebGL, or some engine using it (didn´t Unity have some WebGL export option or something?). WebGL is quite perfect for something like that, the problem is just that it is only fully supported by Chrome and Firefox, last time I checked it was enabable in Safari and supposed to be there in Opera soon, Internet Explorer however won´t have it too soon due to security concerns
Then there also is some drawing functionality with HTML5 which should have wider support already, but it probably isn´t sufficent for more complex model rendering than a few cubes. So what you are left with for now is most probably flash...

As Slin said, browser support is the big issue there.
You can safely bet on Flash to work cross-browser and on all platforms, but Flash does not come with 3D functionality out of the box. You'd need to use something like papervision3D ( http://blog.papervision3d.org/ ).
Unity3D recently has ability to export to flash though, and you can create 3D content through that as well (which I think is the easiest way).

http://www.irrlicht3d.org/pivot/entry.php?id=1298
coppercube could work very well for what you want to do.
it can export to webgl and flash.

there is some solution for also making webgl work on ie. but i forgot its name.

PDF also supports 3D Models. But I guess that would be a rather expensive solution.

Since this is for the development of a website (or website components) for a company, supporting IE (as well as the other browsers) is important. I'm also reluctant to go the Flash route as businesses want their sites fully viewable on the iPad and iPhone as well.

Then you are out of options I fear. You have to use Flash on browsers without WebGL/HTML5 support and switch to WebGL/HTML5 on the others.

Why would you have to use Flash? What I mean is, there used to be a ton of plugin applications out there (like Cult3D) that would do the job. Where did they all go? Isn't there some Java solution?

Now I am even more curious about this ...

Flash is the plugin most people have. Everything else is just installed on a very very very small fraction of PCs. Using something required by Youtube, is a fairly good sign that even your favorite idiot has it installed

I know that. I've been at this for a long time. But I cannot help the wants/needs of a client, you know. If I can offer them a non-Flash solution, they will be all the happier.

In any case, this thread is NOT about the pluses/minuses of Flash or how many people have it installed. It's about solutions for delivering interactive 3D content via the web.