Code:

In[187]:=
u[x_] := Ceiling[(1/2)*(Sqrt[8*x + 1] -
1)];
v[x_] := (1/2)*u[x]*(u[x] + 1) - x;
fold[y_] := {v[y] + 1, u[y] - v[y]};
oldlist = Table[fold[n], {n, 1, 20}]
fnew[y_] := Module[{m, n}, m = v[y] + 1;
n = u[y] - v[y]; If[(-1)^(m + n) == 1,
{m, n}, {n, m}]]
newlist = Table[fnew[n], {n, 1, 20}]
Out[190]=
{{1, 1}, {2, 1}, {1, 2}, {3, 1}, {2, 2},
{1, 3}, {4, 1}, {3, 2}, {2, 3}, {1, 4},
{5, 1}, {4, 2}, {3, 3}, {2, 4}, {1, 5},
{6, 1}, {5, 2}, {4, 3}, {3, 4}, {2, 5}}
Out[192]=
{{1, 1}, {1, 2}, {2, 1}, {3, 1}, {2, 2},
{1, 3}, {1, 4}, {2, 3}, {3, 2}, {4, 1},
{5, 1}, {4, 2}, {3, 3}, {2, 4}, {1, 5},
{1, 6}, {2, 5}, {3, 4}, {4, 3}, {5, 2}}