Q:

Have there ever been simultaneous cyclones in the same ocean but different hemispheres?

I'm teaching a physics class, and I'm getting to the point where we are talking about the Coriolis effect.  I usually show a couple of photos/videos showing a storm in the Northern Hemisphere and a storm in the Southern Hemisphere, and talk about why they rotate different directions. 
However, it occurred to me that at some point in history, there could well have been two storms occurring in the same ocean but in different hemispheres.  Has this ever occurred?  This would most likely have to be in the Pacific or Indian Oceans (probably the Pacific, since from what I've gathered the Northern Indian Ocean is less active than most other cyclone basins.)  Ideally, a photo/video of this event would exist that I could show my students.

A:

It does occasionally happen. Not often, because to kick-start a hurricane there has to be some rotation to start with, from the Coriolis effect, together with a sea surface temperature of >27 °C (+ some other requirements such as lack of horizontal shear). This combination is best met during summer when one or other hemisphere is tilted towards the sun, so the hurricane seasons in both hemispheres tend to be 6 months out of synchronization. But there are occasional overlaps, as in this vector composite, found at http://www.wunderground.com/blog/JeffMasters/archive.html?year=2015&month=06&MR=1

A:

The formation of twin cyclones does occur in both the Indian Ocean as well as the Western Pacific with the Indian Ocean twins being weaker . Schubert et al Symmetric Twin Tropical Cyclones report that the October through May is the period of formation of twin tropical cyclones with December through February being the maximum frequency. The original definition of twin tropical cyclones was provided by Keen in 1982 Cross Equatorial Pairs in the Southern Oscillation. Apart from high SSTs you also need conditions of intense low level  vorticity. The convection associated with the eastward propagating Madden Julien Oscillation(MJO) contains cyclonic Rossby gyres that favour tropical cyclogenesis.   
If students are wondering what and where are the cyclonic gyres this image may help in that regard- please pay attention to the cyclonic gyres at the 850 hPa level that trail the convection

Lander in Formation of Twin Tropical Cyclone Twins Symmetrical With Respect to the Equator  outline the "Keen criteria" 

The northern and southern systems must initially form within 9 days of each other
Latitudinal separation must not be more than 22 degrees
The longitudinal difference of the initial disturbances ranged between +9 degrees and -17 degrees

A further strict definition for twins include

They must form simultaneously
They form at low latitudes (5 degrees above and below the equator) 
They form along the same longitude

Lander performs case studies for the twin cyclones Lola and Namu in the 1986 Pacific Typhoon season - 1986 Pacific Typhoon Season
A more recent reference Schreck et al. A case study of an outbreak of twin tropical cyclones have performed studies that conclude that a broad and long lasting envelope of warm water, low surface pressure and easterly shear are other conditions that are required for formation of twin cyclones. 
Students can watch the video in this URL Cyclone Pam and Bavi and see the twin cyclone activity

A:

To add to Gordon's answer, Cyclone Twenty-five was also called Cyclone Raquel and it was an "off-season" cyclone, as the following reported in the news. It was the first cyclone recorded in the region during July.
The synoptic weather map the region on that day was

Source: http://www.bom.gov.au/cgi-bin/charts/charts.view.pl?idcode=IDX0016&file=IDX0016.201506300000.gif

