Put mango into a large bowl. Sprinkle salt over the mango. Pour water over the mango. Let mango soak for 1 hour to overnight.
Drain and discard liquid from the mango.
Stir sugar and vinegar together in a saucepan over medium heat; bring to a boil. Add drained mango to the liquid; stir to coat. Stir raisins, dates, cinnamon sticks, ginger, garlic, and ground dried chile pepper into the mango mixture; bring again to a boil, stirring occasionally. Reduce heat to low and simmer mixture until thickened, about 1 hour. Remove from heat to cool completely.
Sterilize jars and lids in boiling water for at least 5 minutes. Pack chutney into hot, sterilized jars, filling to within 1/4 inch of the top. Run a knife or thin spatula around the insides of the jars after they have been filled to remove any air bubbles. Wipe the rims of the jars with a moist paper towel to remove any food residue. Top with lids and screw on rings.
Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower jars into the boiling water using a holder. Leave a 2-inch space between the jars. Pour in more boiling water if necessary to bring the water level to at least 1 inch above the tops of the jars. Bring the water to a rolling boil, cover the pot, and process for 15 minutes.
Remove the jars from the stockpot and place onto a cloth-covered or wood surface, several inches apart, until cool. Once cool, press the top of each lid with a finger, ensuring that the seal is tight (lid does not move up or down at all). Store in a cool, dark area.