Is it wrong to space en dashes and em dashes?

That seems the convention. The difference between em and en dashes doesn't matter a great deal; the spacing does, because it makes it much clearer (especially to distinguish dashes – interjecting little notes like this – to a hyphen phrase, for example Anglo-Saxon).

When I was learning typography—many years ago--the convention was that you did NOT use spaces around en- or em-dashes. If the type designer wants extra space for readability or aesthetics, they will build it into the slug for the type (or in more modern cases, program it into the font).

Because of the proliferation of letter-spaced fonts online, where there is no allowance for a typographer to make a character with a little extra space for aesthetics, the logic for the standard doesn't exactly apply anymore.

But that's still how I do it. And as you can see, when dealing with the internet I tend to use double hyphens instead of em-dashes because you never know how a text editor is going to handle an em-dash, and those that handle them at all will often convert a double hyphen into an em-dash.

And as far as I know, substituting a spaced en-dash where you should use an unspaced em-dash is definitely out.

It is technically wrong, but...

I think MS Word contributes to this practice. You cannot get it to automatically make an em-dash from a single hyphen. However, if you are typing along and then type space-dash-space followed by a word - like this - then it will convert the hyphens into an en-dashes. It does not remove the space around them when it does this.

For that reason, I've gotten into the habit of doing it that way when I'm not writing something formal. But the correct practice in Word would be to use '--' and '---' with no surrounding spaces. (Assuming auto-dash is turned on, of course.)