How to determine the direction of the touchmove

To know the difference in horizontal and vertical, you must work on the X and Y axes. This would be so if we were to use the logic of your code.

var ts_x;
var ts_y;
$(document).bind('touchstart', function(e) {
   ts_x = e.originalEvent.touches[0].clientX;
   ts_y = e.originalEvent.touches[0].clientY;
});
$(document).bind('touchend', function(e) {
   var td_x = e.originalEvent.changedTouches[0].clientX - ts_x;
   var td_y = e.originalEvent.changedTouches[0].clientY - ts_y;
   // O movimento principal foi vertical ou horizontal?
   if( Math.abs( td_x ) > Math.abs( td_y ) ) {
      // é horizontal
      if( td_x < 0 ) {
         // esquerda
      } else {
         // direita
      }
   } else {
      // é vertical
      if( td_y < 0 ) {
         // cima
      } else {
         // baixo
      }
 });

A pure JS code follows the original properties of the event, pageX and pageY :

var ts_x;
var ts_y;
document.addEventListener('touchstart', function(e) {
   e.preventDefault();
   var touch = e.changedTouches[0];
   ts_x = touch.pageX;
   ts_y = touch.pageY;
}, false);

document.addEventListener('touchmove', function(e) {
   e.preventDefault();
   var touch = e.changedTouches[0];
   td_x = touch.pageX - ts_x; // deslocamento na horizontal
   td_y = touch.pageY - ts_y; // deslocamento na vertical
   // O movimento principal foi vertical ou horizontal?
   if( Math.abs( td_x ) > Math.abs( td_y ) ) {
      // é horizontal
      if( td_x < 0 ) {
         // é para esquerda
      } else {
         // direita
      }
   } else {
      // é vertical
      if( td_y < 0 ) {
         // cima
      } else {
         // baixo
      }
   }
}, false);

To increment the code, you can check if the movement has passed a certain distance, to ignore touches on the screen by adding something like this:

 if( Math.max( Math.abs( td_x ), Math.abs( td_y ) ) > 10 ...

Note: I used the event touchend instead of touchmove . See which one is most suitable for end use. The end is only triggered when the move is terminated, move can be fired a number of times during dragging.