How to draw tilted axes for spacetime diagrams

Here is a first try. I do not claim that it is perfect, but it may serve as a starting point.:

skeweaxes[{{xmin_, xmax_}, {ymin_, ymax_}}, phi_] := 
 Module[{nxti = 20, nyti = 20, xtlen = (ymax - ymin)/100, 
   ytlen = (xmax - xmin)/100},
  rot0 = RotationTransform[phi]; rot1 = RotationTransform[-phi];
  t0 = {rot0 /@ {{xmin, ymin}, {xmax, ymin}}, 
    rot1 /@ {{xmin, ymin}, {xmin, ymax}}};
  c = 0; t1 = 
   Table[rot0 /@ {{x, ymin}, {x, 
       ymin + xtlen If[Mod[++c, 5] == 1, 2, 1]}}, {x, xmin, 
     xmax, (xmax - xmin)/nxti}];
  c = 0; t2 = 
   Table[rot1 /@ {{xmin, y}, {xmin + ytlen If[Mod[++c, 5] == 1, 2, 1],
        y}}, {y, ymin, ymax, (ymax - ymin)/nyti}];
  
  Graphics[Line[Join[t0, t1, t2]]]
  ]
pl = Plot[{x, x^2}, {x, 0, 1}];
Show[{skeweaxes[{{0, 1}, {0, 1}}, Pi/8], pl}, AspectRatio -> 1, 
 Axes -> True]

Here's an approach based on the fifth argument of Inset:

Graphics[
 {
  Inset[
   Graphics[{},
    Frame -> {{True, False}, {True, False}},
    FrameTicksStyle -> (FontOpacity -> 0),
    PlotRange -> {{0, 1}, {0, 1}}
    ],
   {0, 0}, {0, 0}, {1, 1}, {{1, 0.5}, {0.5, 1}}
   ]
  },
 Frame -> {{True, False}, {True, False}},
 PlotRange -> {{0, 1}, {0, 1}}
 ]

The argument {{1, 0.5}, {0.5, 1}} specifies the direction of the x and y axis of the inner plot respectively. You'll need to compute the exact size the inset needs to be, but other than that there shouldn't be an issue. The other thing to note is that the ticks are also skewed for the tilted coordinate system, but since they point along the direction of the other axis, this might actually be a good thing. Assuming the size and axes directions are correct, it is also trivial to add additional content to the inner graphics and have it displayed in the same warped way as the axes themselves.