Insects flying upside down?  They "attempt" it.
But bees flying upside down.  No.

> Thus dragonflies and damselflies 
> will attempt to fly upside down if 
> illuminated from below.  

Dragonflies are very different from bees, and the phrase "attempt to fly
upside down", seems to indicate that even Dragonflies will not be very
successful in their attempt.

Bees, who fly with their stingers lower than their head, and with their legs
extended, would, once inverted, loose lift, and fall down, not fly forward.
Any forward motion in a "crash landing" after inversion would due to
momentum alone.  The recent film "More Than  Honey" has some great
high-rez/slo-mo video of bees flying, and one can see the bees roll and
pitch as they fly.

The subject has been studied in detail, and one of the biggest factors is
the legs of bees, which turned out to have significant impact on their
aerodynamic performance and are used by the bees to stabilize their flight.

"The bees can reach a maximum speed of 7.25 m/s, but at these speeds they
lose rotational stability: "They roll all the way to the side or often
upside down, and crash to the ground", observes Combes. This means that what
limits the bee's speed is not muscle power or the amplitude of its wing
beat, but the pitch of the body balanced with the resulting rotational
instability. "Having the legs extended generates stabilizing lift forces and
helps reduce the moment of inertia and the slow rolling, similar to when a
spinning figure-skater extends their arms", explains Combes."

http://www.eurekalert.org/pub_releases/2006-04/sfeb-fto033106.php
http://tinyurl.com/lzyaq

And the paper eventually published from the findings:

http://www.pnas.org/content/106/22/9105

"Bees extend their hindlegs ventrally at higher speeds, improving roll
stability but also increasing body drag and associated power requirements by
30%. Despite the energetic cost, we observed this stability-enhancing
behavior in 10 euglossine species from 3 different genera, spanning an order
of magnitude in body size. A field experiment in which we altered the level
of turbulence demonstrates that flight instability and maximum flight speed
are directly related to flow variability."

"Turbulence-driven instabilities limit insect flight performance"
doi: 10.1073/pnas.0902186106

So bees at higher speeds can only maintain control with their legs, as this
allows them to adjust the roll and pitch of their body for the airspeed at
which they are flying (including headwinds, tailwinds, crosswinds).

May and Wakeling did a number of fine reviews of the flight of dragonflies
and damselflies in the 1990s. Very different flight from that of bees, as
they explain in great detail:

http://jeb.biologists.org/content/158/1/325.full.pdf
http://jeb.biologists.org/content/200/3/557.full.pdf‎
http://jeb.biologists.org/content/200/3/583.full.pdf

That said, hummingbirds are capable of brief periods of inverted hover and
flight, but this is a simply matter of the application of raw power.  They
are amazing flying machines. 

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