Can I connect up LED strip lighting at alternate ends?
The LED's are in parallel (for the most part), and thus the + and - nodes are the same at either end of the cable. 

Below is a diagram representing a simple LED strip.



The strip "begins" on the left, and "ends" on the right. This is to say that if you lengthened the strip, you'd simply see the pattern repeat sideways.

Assuming the source is 12V, you can see that the cathode (+) end is 12V, both at the beginning of the strip (left side) and end of the strip (right side).

The same goes for the anode (-). It's 0V at the beginning, and 0V at the other end.

In this simplified example, there is theoretically no difference to grounding your strip at a different end.

Now let's go a bit deeper. First, the strips typically have several LEDs in series with a resistor, and that sequence is repeated in parallel. This method is cheaper and more compact, though perhaps less "accurate" in terms of current balancing. I didn't include that in the first example because it wasn't necessary for understanding the answer to the question.



Finally, realize that actual wires have a bit of resistance. Thus there is a difference to which end you connect the anode vs cathode. But you probably won't notice a difference. If you had a very long LED strip and the resistance added up considerably, you might notice that connecting the anode and cathode at the same end up the strip resulted in the LEDs gradually dimming as they progressed to the other end. This would be because the LEDs closest to that end have the least resistance. The LEDs at the far end would be behind much more wire, and thus have more more resistance. Connecting opposing ends should result in a more balanced brightness across all LEDs.

Below is a simulation that includes resistance between LED segments (exaggerated), and shows the current through each. There's a switch at the bottom that you can click on to switch the ground to either end.
[Link to simulation]