Simplest way to get rbind to ignore column names

You might find setNames handy here...

rbind(df, setNames(rev(df), names(df)))
#  x y
#1 1 3
#2 2 4
#3 3 1
#4 4 2

I suspect your real use-case is somewhat more complex. You can of course reorder columns in the first argument of setNames as you wish, just use names(df) in the second argument, so that the names of the reordered columns match the original.

This seems pretty easy:

mapply(c,df,df[,2:1])
     x y
[1,] 1 3
[2,] 2 4
[3,] 3 1
[4,] 4 2

For this simple case, though, you have to turn it back into a dataframe (because mapply simplifies it to a matrix):

as.data.frame(mapply(c,df,df[,2:1]))
  x y
1 1 3
2 2 4
3 3 1
4 4 2

Important note 1: There appears to be a downside of type coercion when your dataframe contains vectors of different types:

df<-data.frame(x=1:2,y=3:4,z=c('a','b'))
mapply(c,df,df[,c(2:1,3)])
     x y z
[1,] 1 3 2
[2,] 2 4 1
[3,] 3 1 2
[4,] 4 2 1

Important note 2: It also is terrible if you have factors.

df<-data.frame(x=factor(1:2),y=factor(3:4))
mapply(c,df[,1:2],df[,2:1])
     x y
[1,] 1 1
[2,] 2 2
[3,] 1 1
[4,] 2 2

So, as long as you have all numeric data, it's okay.