Harihara

शिवाय विष्णु रूपाय शिव रूपाय विष्णवे |

शिवस्य हृदयं विष्णुं विष्णोश्च हृदयं शिवः ||

यथा शिवमयो विष्णुरेवं विष्णुमयः शिवः |
यथान्तरम न पश्यामि तथा में स्वस्तिरायुषि|
यथान्तरम न भेदा: स्यु: शिवराघवयोस्तथा||
(यजुर्वेद)
Vishnu is Shiva and Shiva is Vishnu. Vishnu resides in Shiva’s heart and Shiva resides in Vishnu’s. Hence, Vishnu can be found in the same place as Shiva and Shiva can be found in the same place as Vishnu. A person who sees them as one and inseparable, is the one to live a gracious life.

यथा शिवस्तथा विष्णुर्यथा विष्णुस्तथा शिव: ।
अन्तरं शिवविष्ण्वोश्र भनागपि न विद्यते ।।
(स्कन्द पुराण)
Where exists Shiva, Vishnu exists with him. And where exists Vishnu, Shiva exists with him. (They both exist in each other and everywhere.) Their eternal existence has no difference.

Harihara, is called the form of supreme god, in which both Adipurushas (birthless eternal beings) exists as one. ‘Hari’ stands for Vishnu and ‘Hara’ stands for Shiva. This particular form is worshipped by all the cults of Hinduism as the supreme form. However, it stands for a deeper meaning. Their eternal existence must be seen as one and infinite. Both Vishnu and Shiva are supreme gods (Swayam Bhagwan). They represent two sides of one coin. Vishnu, the preserver of universe, resides upon the ocean of milk (in which the entire universe cherishes), whereas Shiva, the destroyer of creation, resides on the summit of Divine Kailash (the cosmic mountain situated in the centre of the universe). Where Shiva, despite being an ascetic, has compassion for his family, Vishnu, despite being married, represents an ascetic mind. Shiva, who is married to Adishakti (the very energy of universe), represents how a man can still be egoless and powerful. Vishnu, who is married to Lakshmi (the prosperity of universe), represents how a man can still be generous and without greed in spite of having enormous wealth. Shiva, the destroyer, has a cherishing family. Whereas Vishnu, the preserver, has the universe as his family. Shiva, the founder of Yoga, meditates on Rama’s (Incarnation of Vishnu) blissful name. And Vishnu meditates on the calmness of Shiva. As the Yujarveda says, they both resides in each other’s heart. Vishnu is the only one to calm Shiva’s temper and endure his pain and Shiva is the one to know the endurance and purposes of Vishnu’s incarnations. They live human lives through each other’s heart, establishing the standards of an ideal life. They worship each other in their incarnations. E.g., Hanuman (Incarnation of Shiva) worships Rama (Incarnation of Vishnu) and Rama worships Shiva. They both are each other’s supreme gods and each other’s greatest devotees.
There cannot be any difference between them. The universe is a part of their eternal existence and they both together sustain balance of nature. Their singular existence results into manifestation, preservation and destruction.