As you clean the cobwebs from the corners, you can take comfort in the fact that the ISS has spiders, too. Those are experimental spiders, deliberately taken aboard to see how space conditions affect web-building. In fact, spiders have flown into space for more than ten years, but now it appears there is a breakthrough in our understanding of the way orb spiders build webs in microgravity. From the research paper:
Under natural conditions, Trichonephila spiders build asymmetric webs with the hub near the upper edge of the web, and they always orient themselves downwards when sitting on the hub whilst waiting for prey. As these asymmetries are considered to be linked to gravity, we expected the spiders experiencing no gravity to build symmetric webs and to show a random orientation when sitting on the hub. We found that most, but not all, webs built in zero gravity were indeed quite symmetric. Closer analysis revealed that webs built when the lights were on were more asymmetric (with the hub near the lights) than webs built when the lights were off. In addition, spiders showed a random orientation when the lights were off but faced away from the lights when they were on. We conclude that in the absence of gravity, the direction of light can serve as an orientation guide for spiders during web building and when waiting for prey on the hub.
It appears that in the absence of sufficient gravity, the spiders saw the light source as a substitute for "up." Read a short version of the study's findings, plus a look at previous experiments with spiders in space at Gizmodo.