From what I have read, this is not a problem. If it isn't a problem, then why aren't all plugs able to be plugged in either way?
What you have read is only partially accurate.
In reality, and in most cases, a simple appliance or equipment will normally work fine regardless of which way round the line and neutral is connected. Since the voltage is AC it sort of does not have a polarity like DC has positive and negative terminals.
The issue is the switching. Look at the two circuits below.
simulate this circuit – Schematic created using CircuitLab
The circuit of the left is wired the right way around. When the switch is off the lamp or motor or whatever is connected to neutral and, theoretically, is in a safe state and can be touched without electrocuting you. (Though I would not actually do that just in case the wiring is backwards in general.)
The circuit on the right, on the other hand, has everything live when the switch is open. This makes it a MUCH more dangerous appliance.
Note if the thing contains a fuse, the same issue occurs when the fuse blows. But now, if there happens to be a short to ground in the wiring, the fuse will not blow in the right circuit. Hopefully a ground fault interrupter will take care of that instead.
As such, a standard is employed to ensure the mains is normally connected the right way around.
In reality, in most cases, as long as you do not plan on taking the covers off and working on the guts with the power attached, and the device is a Class II type built correctly with the proper double insulation techniques, the connector orientation does not matter.
Ultimately though you should either use a keyed connector or a non-reversible three pin type and leave the ground pin disconnected.
ADDITION: When cleaning the blades though you should unplug it. Something you should do anyway, but thought I would re-emphasise it. BTW any readers with gas mowers also need to remember to disconnect the spark plug when cleaning the blades too.