Today’s model is a sculpture made with the excellent and innovative mesh-wrangling software TopMod. Since my way-back math roots are in topology, TopMod is like coming home; everything about it speaks my language. Having said that, I’m just a TopMod beginner so my first sculpture is very simple: just six handles arranged around a cubical center and smoothed out with Doo-Sabin. 
STL file: day135_sixhandles.stl Thingiverse link: http://www.thingiverse.com/thing:222160
Settings: Makes a gift-quality print on MakerWare .15mm/high with raft and supports in about 2 hours and 13 minutes.
Technical notes: This model was made in TopMod which you can download for free here. Basic ReadMe and list of operations is here. To make the model I started with a cube and added handles connecting three adjacent pairs of faces. Then I applied the Doo-Sabin mesh subdivision, added three smaller, twisty handles anchored to the new mesh, and applied Doo-Sabin a couple more times.
To get started with TopMod I suggest watching the following YouTube videos:
- GravitoMag’s tutorial of setting up TopMod and making an infinity symbol
- Dave Morris’ (soundless?) tutorial of making twisty cube handles
- Mathieu Glachant’s tutorial of making a beautifully complex shot glass full of holes
- David Brinnen’s tutorial of making a double cross mobius with TopMod and Wings3D
UPDATE: You can now order a print of this model from Shapeways at the new Geekhaus store!
MORE UPDATE: kitwallace has found a way to make this same shape by parametrizing with trigonometric functions.
Thought you might like this alternative 6 handles I stumbled on:
http://kitwallace.tumblr.com/post/74267364600/six-handles